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


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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 b => (a  ->  d  ->  b c)  ->  [a]  ->  [d]  ->  b ();
zipWithM_ f xs ys = sequence_ (zipWith f xs ys);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\_->q"
is transformed to
"gtGt0 q _ = q;
"

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

(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 a => (b  ->  d  ->  a c)  ->  [b]  ->  [d]  ->  a ();
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 b => (c  ->  d  ->  b a)  ->  [c]  ->  [d]  ->  b ();
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 => (b  ->  c  ->  a d)  ->  [b]  ->  [c]  ->  a ();
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_ ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.zipWithM_ ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.zipWithM_ ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence_ (zipWith ww3 ww4 ww5)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr (>>) (return ()) (zipWith ww3 ww4 ww5)",fontsize=16,color="burlywood",shape="triangle"];34[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];7 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 8[label="",style="solid", color="burlywood", weight=3];
35[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) ww5)",fontsize=16,color="burlywood",shape="box"];36[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];8 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 10[label="",style="solid", color="burlywood", weight=3];
37[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 11[label="",style="solid", color="burlywood", weight=3];
9[label="foldr (>>) (return ()) (zipWith ww3 [] ww5)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) (ww50 : ww51))",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3];
11[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) [])",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3];
13[label="foldr (>>) (return ()) (ww3 ww40 ww50 : zipWith ww3 ww41 ww51)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14 -> 12[label="",style="dashed", color="red", weight=0];
14[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];15[label="return ()",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16 -> 18[label="",style="dashed", color="red", weight=0];
16[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="Just ()",fontsize=16,color="green",shape="box"];19 -> 7[label="",style="dashed", color="red", weight=0];
19[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3];
19 -> 21[label="",style="dashed", color="magenta", weight=3];
18[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];18 -> 22[label="",style="solid", color="black", weight=3];
20[label="ww51",fontsize=16,color="green",shape="box"];21[label="ww41",fontsize=16,color="green",shape="box"];22 -> 23[label="",style="dashed", color="red", weight=0];
22[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
24[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];24 -> 29[label="",style="dashed", color="green", weight=3];
24 -> 30[label="",style="dashed", color="green", weight=3];
23[label="ww8 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];38[label="ww8/Nothing",fontsize=10,color="white",style="solid",shape="box"];23 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 27[label="",style="solid", color="burlywood", weight=3];
39[label="ww8/Just ww80",fontsize=10,color="white",style="solid",shape="box"];23 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 28[label="",style="solid", color="burlywood", weight=3];
29[label="ww40",fontsize=16,color="green",shape="box"];30[label="ww50",fontsize=16,color="green",shape="box"];27[label="Nothing >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="Just ww80 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
31[label="Nothing",fontsize=16,color="green",shape="box"];32[label="gtGt0 ww6 ww80",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3];
33[label="ww6",fontsize=16,color="green",shape="box"];}

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

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

   new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, 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_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6


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(10)
YES