/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(10) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM :: Monad d => (a  ->  b  ->  d c)  ->  [a]  ->  [b]  ->  d [c];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\xs->return (x : xs)"
is transformed to
"sequence0 x xs = return (x : xs);
"
The following Lambda expression
"\x->sequence cs >>= sequence0 x"
is transformed to
"sequence1 cs x = sequence cs >>= sequence0 x;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM :: Monad c => (a  ->  b  ->  c d)  ->  [a]  ->  [b]  ->  c [d];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM :: Monad a => (b  ->  c  ->  a d)  ->  [b]  ->  [c]  ->  a [d];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

(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 Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
zipWithM :: Monad a => (c  ->  b  ->  a d)  ->  [c]  ->  [b]  ->  a [d];
zipWithM f xs ys = sequence (zipWith f xs ys);

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.zipWithM wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.zipWithM wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.zipWithM wv3 wv4 wv5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (zipWith wv3 wv4 wv5)",fontsize=16,color="burlywood",shape="triangle"];38[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];6 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 7[label="",style="solid", color="burlywood", weight=3];
39[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="sequence (zipWith wv3 (wv40 : wv41) wv5)",fontsize=16,color="burlywood",shape="box"];40[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];7 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 9[label="",style="solid", color="burlywood", weight=3];
41[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="sequence (zipWith wv3 [] wv5)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="sequence (zipWith wv3 (wv40 : wv41) (wv50 : wv51))",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="sequence (zipWith wv3 (wv40 : wv41) [])",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="sequence []",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="sequence (wv3 wv40 wv50 : zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13 -> 11[label="",style="dashed", color="red", weight=0];
13[label="sequence []",fontsize=16,color="magenta"];14[label="return []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15 -> 17[label="",style="dashed", color="red", weight=0];
15[label="wv3 wv40 wv50 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];15 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="Just []",fontsize=16,color="green",shape="box"];18[label="wv3 wv40 wv50",fontsize=16,color="green",shape="box"];18 -> 23[label="",style="dashed", color="green", weight=3];
18 -> 24[label="",style="dashed", color="green", weight=3];
17[label="wv6 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="burlywood",shape="triangle"];42[label="wv6/Nothing",fontsize=10,color="white",style="solid",shape="box"];17 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 21[label="",style="solid", color="burlywood", weight=3];
43[label="wv6/Just wv60",fontsize=10,color="white",style="solid",shape="box"];17 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 22[label="",style="solid", color="burlywood", weight=3];
23[label="wv40",fontsize=16,color="green",shape="box"];24[label="wv50",fontsize=16,color="green",shape="box"];21[label="Nothing >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="Just wv60 >>= sequence1 (zipWith wv3 wv41 wv51)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
25[label="Nothing",fontsize=16,color="green",shape="box"];26[label="sequence1 (zipWith wv3 wv41 wv51) wv60",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
27 -> 28[label="",style="dashed", color="red", weight=0];
27[label="sequence (zipWith wv3 wv41 wv51) >>= sequence0 wv60",fontsize=16,color="magenta"];27 -> 29[label="",style="dashed", color="magenta", weight=3];
29 -> 6[label="",style="dashed", color="red", weight=0];
29[label="sequence (zipWith wv3 wv41 wv51)",fontsize=16,color="magenta"];29 -> 30[label="",style="dashed", color="magenta", weight=3];
29 -> 31[label="",style="dashed", color="magenta", weight=3];
28[label="wv7 >>= sequence0 wv60",fontsize=16,color="burlywood",shape="triangle"];44[label="wv7/Nothing",fontsize=10,color="white",style="solid",shape="box"];28 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 32[label="",style="solid", color="burlywood", weight=3];
45[label="wv7/Just wv70",fontsize=10,color="white",style="solid",shape="box"];28 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 33[label="",style="solid", color="burlywood", weight=3];
30[label="wv41",fontsize=16,color="green",shape="box"];31[label="wv51",fontsize=16,color="green",shape="box"];32[label="Nothing >>= sequence0 wv60",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="Just wv70 >>= sequence0 wv60",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="Nothing",fontsize=16,color="green",shape="box"];35[label="sequence0 wv60 wv70",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
36[label="return (wv60 : wv70)",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
37[label="Just (wv60 : wv70)",fontsize=16,color="green",shape="box"];}

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, h, ba, bb)
   new_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) -> new_gtGtEs(wv3, wv41, wv51, h, ba, bb)

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

(9) 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_sequence(wv3, :(wv40, wv41), :(wv50, wv51), h, ba, bb) -> new_gtGtEs(wv3, wv41, wv51, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6


*new_gtGtEs(wv3, wv41, wv51, h, ba, bb) -> new_sequence(wv3, wv41, wv51, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6


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

(10)
YES