/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Main where {
import qualified Prelude;
import qualified Queue;
}
module Queue where {
import qualified Main;
import qualified Prelude;
data Queue a = Q [a] [a] [a] ;

deQueue :: Queue a  ->  Maybe (a,Queue a);
deQueue (Q [] _ _) = Nothing;
deQueue (Q (x : xs) ys xs') = Just (x,makeQ xs ys xs');

listToQueue :: [a]  ->  Queue a;
listToQueue xs = Q xs [] xs;

makeQ :: [a]  ->  [a]  ->  [a]  ->  Queue a;
makeQ xs ys [] = listToQueue (rotate xs ys []);
makeQ xs ys (_ : xs') = Q xs ys xs';

rotate :: [a]  ->  [a]  ->  [a]  ->  [a];
rotate [] (y : _) zs = y : zs;
rotate (x : xs) (y : ys) zs = x : rotate xs ys (y : zs);

}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
import qualified Queue;
}
module Queue where {
import qualified Main;
import qualified Prelude;
data Queue a = Q [a] [a] [a] ;

deQueue :: Queue a  ->  Maybe (a,Queue a);
deQueue (Q [] vy vz) = Nothing;
deQueue (Q (x : xs) ys xs') = Just (x,makeQ xs ys xs');

listToQueue :: [a]  ->  Queue a;
listToQueue xs = Q xs [] xs;

makeQ :: [a]  ->  [a]  ->  [a]  ->  Queue a;
makeQ xs ys [] = listToQueue (rotate xs ys []);
makeQ xs ys (wu : xs') = Q xs ys xs';

rotate :: [a]  ->  [a]  ->  [a]  ->  [a];
rotate [] (y : vx) zs = y : zs;
rotate (x : xs) (y : ys) zs = x : rotate xs ys (y : zs);

}

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

(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 Main where {
import qualified Prelude;
import qualified Queue;
}
module Queue where {
import qualified Main;
import qualified Prelude;
data Queue a = Q [a] [a] [a] ;

deQueue :: Queue a  ->  Maybe (a,Queue a);
deQueue (Q [] vy vz) = Nothing;
deQueue (Q (x : xs) ys xs') = Just (x,makeQ xs ys xs');

listToQueue :: [a]  ->  Queue a;
listToQueue xs = Q xs [] xs;

makeQ :: [a]  ->  [a]  ->  [a]  ->  Queue a;
makeQ xs ys [] = listToQueue (rotate xs ys []);
makeQ xs ys (wu : xs') = Q xs ys xs';

rotate :: [a]  ->  [a]  ->  [a]  ->  [a];
rotate [] (y : vx) zs = y : zs;
rotate (x : xs) (y : ys) zs = x : rotate xs ys (y : zs);

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Queue.deQueue",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Queue.deQueue wv3",fontsize=16,color="burlywood",shape="triangle"];168[label="wv3/Queue.Q wv30 wv31 wv32",fontsize=10,color="white",style="solid",shape="box"];3 -> 168[label="",style="solid", color="burlywood", weight=9];
168 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="Queue.deQueue (Queue.Q wv30 wv31 wv32)",fontsize=16,color="burlywood",shape="box"];169[label="wv30/wv300 : wv301",fontsize=10,color="white",style="solid",shape="box"];4 -> 169[label="",style="solid", color="burlywood", weight=9];
169 -> 5[label="",style="solid", color="burlywood", weight=3];
170[label="wv30/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 170[label="",style="solid", color="burlywood", weight=9];
170 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="Queue.deQueue (Queue.Q (wv300 : wv301) wv31 wv32)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="Queue.deQueue (Queue.Q [] wv31 wv32)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="Just (wv300,Queue.makeQ wv301 wv31 wv32)",fontsize=16,color="green",shape="box"];7 -> 9[label="",style="dashed", color="green", weight=3];
8[label="Nothing",fontsize=16,color="green",shape="box"];9[label="Queue.makeQ wv301 wv31 wv32",fontsize=16,color="burlywood",shape="box"];171[label="wv32/wv320 : wv321",fontsize=10,color="white",style="solid",shape="box"];9 -> 171[label="",style="solid", color="burlywood", weight=9];
171 -> 10[label="",style="solid", color="burlywood", weight=3];
172[label="wv32/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 172[label="",style="solid", color="burlywood", weight=9];
172 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="Queue.makeQ wv301 wv31 (wv320 : wv321)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="Queue.makeQ wv301 wv31 []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="Queue.Q wv301 wv31 wv321",fontsize=16,color="green",shape="box"];13[label="Queue.listToQueue (Queue.rotate wv301 wv31 [])",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="Queue.Q (Queue.rotate wv301 wv31 []) [] (Queue.rotate wv301 wv31 [])",fontsize=16,color="green",shape="box"];14 -> 15[label="",style="dashed", color="green", weight=3];
14 -> 16[label="",style="dashed", color="green", weight=3];
15[label="Queue.rotate wv301 wv31 []",fontsize=16,color="burlywood",shape="triangle"];173[label="wv301/wv3010 : wv3011",fontsize=10,color="white",style="solid",shape="box"];15 -> 173[label="",style="solid", color="burlywood", weight=9];
173 -> 17[label="",style="solid", color="burlywood", weight=3];
174[label="wv301/[]",fontsize=10,color="white",style="solid",shape="box"];15 -> 174[label="",style="solid", color="burlywood", weight=9];
174 -> 18[label="",style="solid", color="burlywood", weight=3];
16 -> 15[label="",style="dashed", color="red", weight=0];
16[label="Queue.rotate wv301 wv31 []",fontsize=16,color="magenta"];17[label="Queue.rotate (wv3010 : wv3011) wv31 []",fontsize=16,color="burlywood",shape="box"];175[label="wv31/wv310 : wv311",fontsize=10,color="white",style="solid",shape="box"];17 -> 175[label="",style="solid", color="burlywood", weight=9];
175 -> 19[label="",style="solid", color="burlywood", weight=3];
176[label="wv31/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 176[label="",style="solid", color="burlywood", weight=9];
176 -> 20[label="",style="solid", color="burlywood", weight=3];
18[label="Queue.rotate [] wv31 []",fontsize=16,color="burlywood",shape="box"];177[label="wv31/wv310 : wv311",fontsize=10,color="white",style="solid",shape="box"];18 -> 177[label="",style="solid", color="burlywood", weight=9];
177 -> 21[label="",style="solid", color="burlywood", weight=3];
178[label="wv31/[]",fontsize=10,color="white",style="solid",shape="box"];18 -> 178[label="",style="solid", color="burlywood", weight=9];
178 -> 22[label="",style="solid", color="burlywood", weight=3];
19[label="Queue.rotate (wv3010 : wv3011) (wv310 : wv311) []",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="Queue.rotate (wv3010 : wv3011) [] []",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="Queue.rotate [] (wv310 : wv311) []",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="Queue.rotate [] [] []",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="wv3010 : Queue.rotate wv3011 wv311 (wv310 : [])",fontsize=16,color="green",shape="box"];23 -> 27[label="",style="dashed", color="green", weight=3];
24[label="error []",fontsize=16,color="red",shape="box"];25[label="wv310 : []",fontsize=16,color="green",shape="box"];26[label="error []",fontsize=16,color="red",shape="box"];27 -> 116[label="",style="dashed", color="red", weight=0];
27[label="Queue.rotate wv3011 wv311 (wv310 : [])",fontsize=16,color="magenta"];27 -> 117[label="",style="dashed", color="magenta", weight=3];
27 -> 118[label="",style="dashed", color="magenta", weight=3];
27 -> 119[label="",style="dashed", color="magenta", weight=3];
27 -> 120[label="",style="dashed", color="magenta", weight=3];
117[label="wv3011",fontsize=16,color="green",shape="box"];118[label="wv310",fontsize=16,color="green",shape="box"];119[label="[]",fontsize=16,color="green",shape="box"];120[label="wv311",fontsize=16,color="green",shape="box"];116[label="Queue.rotate wv5 wv6 (wv7 : wv8)",fontsize=16,color="burlywood",shape="triangle"];179[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];116 -> 179[label="",style="solid", color="burlywood", weight=9];
179 -> 153[label="",style="solid", color="burlywood", weight=3];
180[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];116 -> 180[label="",style="solid", color="burlywood", weight=9];
180 -> 154[label="",style="solid", color="burlywood", weight=3];
153[label="Queue.rotate (wv50 : wv51) wv6 (wv7 : wv8)",fontsize=16,color="burlywood",shape="box"];181[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];153 -> 181[label="",style="solid", color="burlywood", weight=9];
181 -> 155[label="",style="solid", color="burlywood", weight=3];
182[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];153 -> 182[label="",style="solid", color="burlywood", weight=9];
182 -> 156[label="",style="solid", color="burlywood", weight=3];
154[label="Queue.rotate [] wv6 (wv7 : wv8)",fontsize=16,color="burlywood",shape="box"];183[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];154 -> 183[label="",style="solid", color="burlywood", weight=9];
183 -> 157[label="",style="solid", color="burlywood", weight=3];
184[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];154 -> 184[label="",style="solid", color="burlywood", weight=9];
184 -> 158[label="",style="solid", color="burlywood", weight=3];
155[label="Queue.rotate (wv50 : wv51) (wv60 : wv61) (wv7 : wv8)",fontsize=16,color="black",shape="box"];155 -> 159[label="",style="solid", color="black", weight=3];
156[label="Queue.rotate (wv50 : wv51) [] (wv7 : wv8)",fontsize=16,color="black",shape="box"];156 -> 160[label="",style="solid", color="black", weight=3];
157[label="Queue.rotate [] (wv60 : wv61) (wv7 : wv8)",fontsize=16,color="black",shape="box"];157 -> 161[label="",style="solid", color="black", weight=3];
158[label="Queue.rotate [] [] (wv7 : wv8)",fontsize=16,color="black",shape="box"];158 -> 162[label="",style="solid", color="black", weight=3];
159[label="wv50 : Queue.rotate wv51 wv61 (wv60 : wv7 : wv8)",fontsize=16,color="green",shape="box"];159 -> 163[label="",style="dashed", color="green", weight=3];
160[label="error []",fontsize=16,color="red",shape="box"];161[label="wv60 : wv7 : wv8",fontsize=16,color="green",shape="box"];162[label="error []",fontsize=16,color="red",shape="box"];163 -> 116[label="",style="dashed", color="red", weight=0];
163[label="Queue.rotate wv51 wv61 (wv60 : wv7 : wv8)",fontsize=16,color="magenta"];163 -> 164[label="",style="dashed", color="magenta", weight=3];
163 -> 165[label="",style="dashed", color="magenta", weight=3];
163 -> 166[label="",style="dashed", color="magenta", weight=3];
163 -> 167[label="",style="dashed", color="magenta", weight=3];
164[label="wv51",fontsize=16,color="green",shape="box"];165[label="wv60",fontsize=16,color="green",shape="box"];166[label="wv7 : wv8",fontsize=16,color="green",shape="box"];167[label="wv61",fontsize=16,color="green",shape="box"];}

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

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

   new_rotate(:(wv50, wv51), :(wv60, wv61), wv7, wv8, h) -> new_rotate(wv51, wv61, wv60, :(wv7, wv8), h)

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_rotate(:(wv50, wv51), :(wv60, wv61), wv7, wv8, h) -> new_rotate(wv51, wv61, wv60, :(wv7, wv8), h)
The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 5 >= 5


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

(8)
YES