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


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


YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(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;
mapAccumR :: (a  ->  c  ->  (a,b))  ->  a  ->  [c]  ->  (a,[b]);
mapAccumR _ s [] = (s,[]);
mapAccumR f s (x : xs) = (s'',y : ys) where { 
s' = (\(s',_) ->s') vv8;
s'' = (\(s'',_) ->s'') vv7;
vv7 = f s' x;
vv8 = mapAccumR f s xs;
y = (\(_,y) ->y) vv7;
ys = (\(_,ys) ->ys) vv8;
 };

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

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\(s'',_)->s''"
is transformed to
"s''0 (s'',_) = s'';
"
The following Lambda expression
"\(s',_)->s'"
is transformed to
"s'0 (s',_) = s';
"
The following Lambda expression
"\(_,y)->y"
is transformed to
"y0 (_,y) = y;
"
The following Lambda expression
"\(_,ys)->ys"
is transformed to
"ys0 (_,ys) = 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;
mapAccumR :: (c  ->  a  ->  (c,b))  ->  c  ->  [a]  ->  (c,[b]);
mapAccumR _ s [] = (s,[]);
mapAccumR f s (x : xs) = (s'',y : ys) where { 
s' = s'0 vv8;
s'' = s''0 vv7;
s''0 (s'',_) = s'';
s'0 (s',_) = s';
vv7 = f s' x;
vv8 = mapAccumR f s xs;
y = y0 vv7;
y0 (_,y) = y;
ys = ys0 vv8;
ys0 (_,ys) = 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.
----------------------------------------

(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;
mapAccumR :: (b  ->  a  ->  (b,c))  ->  b  ->  [a]  ->  (b,[c]);
mapAccumR vy s [] = (s,[]);
mapAccumR f s (x : xs) = (s'',y : ys) where { 
s' = s'0 vv8;
s'' = s''0 vv7;
s''0 (s'',wv) = s'';
s'0 (s',ww) = s';
vv7 = f s' x;
vv8 = mapAccumR f s xs;
y = y0 vv7;
y0 (vz,y) = y;
ys = ys0 vv8;
ys0 (wu,ys) = 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;
mapAccumR :: (c  ->  b  ->  (c,a))  ->  c  ->  [b]  ->  (c,[a]);
mapAccumR vy s [] = (s,[]);
mapAccumR f s (x : xs) = (s'',y : ys) where { 
s' = s'0 vv8;
s'' = s''0 vv7;
s''0 (s'',wv) = s'';
s'0 (s',ww) = s';
vv7 = f s' x;
vv8 = mapAccumR f s xs;
y = y0 vv7;
y0 (vz,y) = y;
ys = ys0 vv8;
ys0 (wu,ys) = ys;
 };

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

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

(7) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"(s'',y : ys) where {
s'  = s'0 vv8;
;
s''  = s''0 vv7;
;
s''0 (s'',wv) = s'';
;
s'0 (s',ww) = s';
;
vv7  = f s' x;
;
vv8  = mapAccumR f s xs;
;
y  = y0 vv7;
;
y0 (vz,y) = y;
;
ys  = ys0 vv8;
;
ys0 (wu,ys) = ys;
} 
"
are unpacked to the following functions on top level
"mapAccumRY0 wx wy wz xu (vz,y) = y;
"
"mapAccumRVv8 wx wy wz xu = mapAccumR wx wy wz;
"
"mapAccumRVv7 wx wy wz xu = wx (mapAccumRS' wx wy wz xu) xu;
"
"mapAccumRS''0 wx wy wz xu (s'',wv) = s'';
"
"mapAccumRYs wx wy wz xu = mapAccumRYs0 wx wy wz xu (mapAccumRVv8 wx wy wz xu);
"
"mapAccumRY wx wy wz xu = mapAccumRY0 wx wy wz xu (mapAccumRVv7 wx wy wz xu);
"
"mapAccumRS' wx wy wz xu = mapAccumRS'0 wx wy wz xu (mapAccumRVv8 wx wy wz xu);
"
"mapAccumRS'' wx wy wz xu = mapAccumRS''0 wx wy wz xu (mapAccumRVv7 wx wy wz xu);
"
"mapAccumRYs0 wx wy wz xu (wu,ys) = ys;
"
"mapAccumRS'0 wx wy wz xu (s',ww) = s';
"

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

(8)
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;
mapAccumR :: (c  ->  b  ->  (c,a))  ->  c  ->  [b]  ->  (c,[a]);
mapAccumR vy s [] = (s,[]);
mapAccumR f s (x : xs) = (mapAccumRS'' f s xs x,mapAccumRY f s xs x : mapAccumRYs f s xs x);

mapAccumRS' wx wy wz xu = mapAccumRS'0 wx wy wz xu (mapAccumRVv8 wx wy wz xu);

mapAccumRS'' wx wy wz xu = mapAccumRS''0 wx wy wz xu (mapAccumRVv7 wx wy wz xu);

mapAccumRS''0 wx wy wz xu (s'',wv) = s'';

mapAccumRS'0 wx wy wz xu (s',ww) = s';

mapAccumRVv7 wx wy wz xu = wx (mapAccumRS' wx wy wz xu) xu;

mapAccumRVv8 wx wy wz xu = mapAccumR wx wy wz;

mapAccumRY wx wy wz xu = mapAccumRY0 wx wy wz xu (mapAccumRVv7 wx wy wz xu);

mapAccumRY0 wx wy wz xu (vz,y) = y;

mapAccumRYs wx wy wz xu = mapAccumRYs0 wx wy wz xu (mapAccumRVv8 wx wy wz xu);

mapAccumRYs0 wx wy wz xu (wu,ys) = ys;

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

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

(9) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.mapAccumR",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.mapAccumR xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.mapAccumR xv3 xv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="List.mapAccumR xv3 xv4 xv5",fontsize=16,color="burlywood",shape="triangle"];44[label="xv5/xv50 : xv51",fontsize=10,color="white",style="solid",shape="box"];5 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 6[label="",style="solid", color="burlywood", weight=3];
45[label="xv5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="List.mapAccumR xv3 xv4 (xv50 : xv51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="List.mapAccumR xv3 xv4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="(List.mapAccumRS'' xv3 xv4 xv51 xv50,List.mapAccumRY xv3 xv4 xv51 xv50 : List.mapAccumRYs xv3 xv4 xv51 xv50)",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
8 -> 11[label="",style="dashed", color="green", weight=3];
8 -> 12[label="",style="dashed", color="green", weight=3];
9[label="(xv4,[])",fontsize=16,color="green",shape="box"];10[label="List.mapAccumRS'' xv3 xv4 xv51 xv50",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="List.mapAccumRY xv3 xv4 xv51 xv50",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="List.mapAccumRYs xv3 xv4 xv51 xv50",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13 -> 19[label="",style="dashed", color="red", weight=0];
13[label="List.mapAccumRS''0 xv3 xv4 xv51 xv50 (List.mapAccumRVv7 xv3 xv4 xv51 xv50)",fontsize=16,color="magenta"];13 -> 20[label="",style="dashed", color="magenta", weight=3];
14 -> 24[label="",style="dashed", color="red", weight=0];
14[label="List.mapAccumRY0 xv3 xv4 xv51 xv50 (List.mapAccumRVv7 xv3 xv4 xv51 xv50)",fontsize=16,color="magenta"];14 -> 25[label="",style="dashed", color="magenta", weight=3];
15 -> 28[label="",style="dashed", color="red", weight=0];
15[label="List.mapAccumRYs0 xv3 xv4 xv51 xv50 (List.mapAccumRVv8 xv3 xv4 xv51 xv50)",fontsize=16,color="magenta"];15 -> 29[label="",style="dashed", color="magenta", weight=3];
20[label="List.mapAccumRVv7 xv3 xv4 xv51 xv50",fontsize=16,color="black",shape="triangle"];20 -> 22[label="",style="solid", color="black", weight=3];
19[label="List.mapAccumRS''0 xv3 xv4 xv51 xv50 xv6",fontsize=16,color="burlywood",shape="triangle"];46[label="xv6/(xv60,xv61)",fontsize=10,color="white",style="solid",shape="box"];19 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 23[label="",style="solid", color="burlywood", weight=3];
25 -> 20[label="",style="dashed", color="red", weight=0];
25[label="List.mapAccumRVv7 xv3 xv4 xv51 xv50",fontsize=16,color="magenta"];24[label="List.mapAccumRY0 xv3 xv4 xv51 xv50 xv7",fontsize=16,color="burlywood",shape="triangle"];47[label="xv7/(xv70,xv71)",fontsize=10,color="white",style="solid",shape="box"];24 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 27[label="",style="solid", color="burlywood", weight=3];
29[label="List.mapAccumRVv8 xv3 xv4 xv51 xv50",fontsize=16,color="black",shape="triangle"];29 -> 31[label="",style="solid", color="black", weight=3];
28[label="List.mapAccumRYs0 xv3 xv4 xv51 xv50 xv8",fontsize=16,color="burlywood",shape="triangle"];48[label="xv8/(xv80,xv81)",fontsize=10,color="white",style="solid",shape="box"];28 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 32[label="",style="solid", color="burlywood", weight=3];
22[label="xv3 (List.mapAccumRS' xv3 xv4 xv51 xv50) xv50",fontsize=16,color="green",shape="box"];22 -> 33[label="",style="dashed", color="green", weight=3];
22 -> 34[label="",style="dashed", color="green", weight=3];
23[label="List.mapAccumRS''0 xv3 xv4 xv51 xv50 (xv60,xv61)",fontsize=16,color="black",shape="box"];23 -> 35[label="",style="solid", color="black", weight=3];
27[label="List.mapAccumRY0 xv3 xv4 xv51 xv50 (xv70,xv71)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
31 -> 5[label="",style="dashed", color="red", weight=0];
31[label="List.mapAccumR xv3 xv4 xv51",fontsize=16,color="magenta"];31 -> 37[label="",style="dashed", color="magenta", weight=3];
32[label="List.mapAccumRYs0 xv3 xv4 xv51 xv50 (xv80,xv81)",fontsize=16,color="black",shape="box"];32 -> 38[label="",style="solid", color="black", weight=3];
33[label="List.mapAccumRS' xv3 xv4 xv51 xv50",fontsize=16,color="black",shape="box"];33 -> 39[label="",style="solid", color="black", weight=3];
34[label="xv50",fontsize=16,color="green",shape="box"];35[label="xv60",fontsize=16,color="green",shape="box"];36[label="xv71",fontsize=16,color="green",shape="box"];37[label="xv51",fontsize=16,color="green",shape="box"];38[label="xv81",fontsize=16,color="green",shape="box"];39 -> 40[label="",style="dashed", color="red", weight=0];
39[label="List.mapAccumRS'0 xv3 xv4 xv51 xv50 (List.mapAccumRVv8 xv3 xv4 xv51 xv50)",fontsize=16,color="magenta"];39 -> 41[label="",style="dashed", color="magenta", weight=3];
41 -> 29[label="",style="dashed", color="red", weight=0];
41[label="List.mapAccumRVv8 xv3 xv4 xv51 xv50",fontsize=16,color="magenta"];40[label="List.mapAccumRS'0 xv3 xv4 xv51 xv50 xv9",fontsize=16,color="burlywood",shape="triangle"];49[label="xv9/(xv90,xv91)",fontsize=10,color="white",style="solid",shape="box"];40 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 42[label="",style="solid", color="burlywood", weight=3];
42[label="List.mapAccumRS'0 xv3 xv4 xv51 xv50 (xv90,xv91)",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
43[label="xv90",fontsize=16,color="green",shape="box"];}

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

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

   new_mapAccumRVv8(xv3, xv4, xv51, xv50, ba, bb, bc) -> new_mapAccumR(xv3, xv4, xv51, ba, bb, bc)
   new_mapAccumR(xv3, xv4, :(xv50, xv51), ba, bb, bc) -> new_mapAccumRVv8(xv3, xv4, xv51, xv50, ba, bb, bc)
   new_mapAccumR(xv3, xv4, :(xv50, xv51), ba, bb, bc) -> new_mapAccumRVv7(xv3, xv4, xv51, xv50, ba, bb, bc)
   new_mapAccumRVv7(xv3, xv4, xv51, xv50, ba, bb, bc) -> new_mapAccumRVv8(xv3, xv4, xv51, xv50, ba, bb, bc)
   new_mapAccumR(xv3, xv4, :(xv50, xv51), ba, bb, bc) -> new_mapAccumR(xv3, xv4, xv51, ba, bb, bc)

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_mapAccumR(xv3, xv4, :(xv50, xv51), ba, bb, bc) -> new_mapAccumRVv8(xv3, xv4, xv51, xv50, ba, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5, 5 >= 6, 6 >= 7


*new_mapAccumRVv7(xv3, xv4, xv51, xv50, ba, bb, bc) -> new_mapAccumRVv8(xv3, xv4, xv51, xv50, ba, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7


*new_mapAccumRVv8(xv3, xv4, xv51, xv50, ba, bb, bc) -> new_mapAccumR(xv3, xv4, xv51, ba, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 5 >= 4, 6 >= 5, 7 >= 6


*new_mapAccumR(xv3, xv4, :(xv50, xv51), ba, bb, bc) -> new_mapAccumR(xv3, xv4, xv51, ba, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6


*new_mapAccumR(xv3, xv4, :(xv50, xv51), ba, bb, bc) -> new_mapAccumRVv7(xv3, xv4, xv51, xv50, ba, bb, bc)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5, 5 >= 6, 6 >= 7


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

(12)
YES