/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


MAYBE
proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could not be shown:

(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) AND
    (9) QDP
        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
        (11) AND
            (12) QDP
                (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (14) YES
            (15) QDP
                (16) MRRProof [EQUIVALENT, 0 ms]
                (17) QDP
                (18) DependencyGraphProof [EQUIVALENT, 0 ms]
                (19) QDP
                (20) NonTerminationLoopProof [COMPLETE, 3 ms]
                (21) NO
            (22) QDP
                (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (24) YES
    (25) QDP
        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (27) YES
(28) Narrow [COMPLETE, 0 ms]
(29) QDP
    (30) DependencyGraphProof [EQUIVALENT, 0 ms]
    (31) TRUE


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

(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;
foldM :: Monad b => (a  ->  c  ->  b a)  ->  a  ->  [c]  ->  b a;
foldM _ a [] = return a;
foldM f a (x : xs) = f a x >>= (\fax ->foldM f fax xs);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\fax->foldM f fax xs"
is transformed to
"foldM0 f xs fax = foldM f fax xs;
"

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

(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;
foldM :: Monad a => (b  ->  c  ->  a b)  ->  b  ->  [c]  ->  a b;
foldM _ a [] = return a;
foldM f a (x : xs) = f a x >>= foldM0 f xs;

foldM0 f xs fax = foldM f fax xs;

}

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

(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;
foldM :: Monad b => (a  ->  c  ->  b a)  ->  a  ->  [c]  ->  b a;
foldM vy a [] = return a;
foldM f a (x : xs) = f a x >>= foldM0 f xs;

foldM0 f xs fax = foldM f fax xs;

}

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

(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;
foldM :: Monad a => (b  ->  c  ->  a b)  ->  b  ->  [c]  ->  a b;
foldM vy a [] = return a;
foldM f a (x : xs) = f a x >>= foldM0 f xs;

foldM0 f xs fax = foldM f fax xs;

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.foldM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.foldM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.foldM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.foldM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];76[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 6[label="",style="solid", color="burlywood", weight=3];
77[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="Monad.foldM vz3 vz4 (vz50 : vz51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="Monad.foldM vz3 vz4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="blue",shape="box"];78[label=">>= :: (IO a) -> (a -> IO a) -> IO a",fontsize=10,color="white",style="solid",shape="box"];8 -> 78[label="",style="solid", color="blue", weight=9];
78 -> 10[label="",style="solid", color="blue", weight=3];
79[label=">>= :: ([] a) -> (a -> [] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];8 -> 79[label="",style="solid", color="blue", weight=9];
79 -> 11[label="",style="solid", color="blue", weight=3];
80[label=">>= :: (Maybe a) -> (a -> Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];8 -> 80[label="",style="solid", color="blue", weight=9];
80 -> 12[label="",style="solid", color="blue", weight=3];
9[label="return vz4",fontsize=16,color="blue",shape="box"];81[label="return :: a -> IO a",fontsize=10,color="white",style="solid",shape="box"];9 -> 81[label="",style="solid", color="blue", weight=9];
81 -> 13[label="",style="solid", color="blue", weight=3];
82[label="return :: a -> [] a",fontsize=10,color="white",style="solid",shape="box"];9 -> 82[label="",style="solid", color="blue", weight=9];
82 -> 14[label="",style="solid", color="blue", weight=3];
83[label="return :: a -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];9 -> 83[label="",style="solid", color="blue", weight=9];
83 -> 15[label="",style="solid", color="blue", weight=3];
10[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];10 -> 16[label="",style="solid", color="black", weight=3];
11 -> 17[label="",style="dashed", color="red", weight=0];
11[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];11 -> 18[label="",style="dashed", color="magenta", weight=3];
12 -> 19[label="",style="dashed", color="red", weight=0];
12[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];12 -> 20[label="",style="dashed", color="magenta", weight=3];
13[label="return vz4",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="return vz4",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="return vz4",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16 -> 24[label="",style="dashed", color="red", weight=0];
16[label="primbindIO (vz3 vz4 vz50) (Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];16 -> 25[label="",style="dashed", color="magenta", weight=3];
18[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];18 -> 26[label="",style="dashed", color="green", weight=3];
18 -> 27[label="",style="dashed", color="green", weight=3];
17[label="vz6 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];84[label="vz6/vz60 : vz61",fontsize=10,color="white",style="solid",shape="box"];17 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 28[label="",style="solid", color="burlywood", weight=3];
85[label="vz6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 29[label="",style="solid", color="burlywood", weight=3];
20[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];20 -> 30[label="",style="dashed", color="green", weight=3];
20 -> 31[label="",style="dashed", color="green", weight=3];
19[label="vz7 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];86[label="vz7/Nothing",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 32[label="",style="solid", color="burlywood", weight=3];
87[label="vz7/Just vz70",fontsize=10,color="white",style="solid",shape="box"];19 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 33[label="",style="solid", color="burlywood", weight=3];
21[label="primretIO vz4",fontsize=16,color="black",shape="box"];21 -> 34[label="",style="solid", color="black", weight=3];
22[label="vz4 : []",fontsize=16,color="green",shape="box"];23[label="Just vz4",fontsize=16,color="green",shape="box"];25[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];25 -> 41[label="",style="dashed", color="green", weight=3];
25 -> 42[label="",style="dashed", color="green", weight=3];
24[label="primbindIO vz8 (Monad.foldM0 vz3 vz51)",fontsize=16,color="burlywood",shape="triangle"];88[label="vz8/IO vz80",fontsize=10,color="white",style="solid",shape="box"];24 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 37[label="",style="solid", color="burlywood", weight=3];
89[label="vz8/AProVE_IO vz80",fontsize=10,color="white",style="solid",shape="box"];24 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 38[label="",style="solid", color="burlywood", weight=3];
90[label="vz8/AProVE_Exception vz80",fontsize=10,color="white",style="solid",shape="box"];24 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 39[label="",style="solid", color="burlywood", weight=3];
91[label="vz8/AProVE_Error vz80",fontsize=10,color="white",style="solid",shape="box"];24 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 40[label="",style="solid", color="burlywood", weight=3];
26[label="vz4",fontsize=16,color="green",shape="box"];27[label="vz50",fontsize=16,color="green",shape="box"];28[label="vz60 : vz61 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];28 -> 43[label="",style="solid", color="black", weight=3];
29[label="[] >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];29 -> 44[label="",style="solid", color="black", weight=3];
30[label="vz4",fontsize=16,color="green",shape="box"];31[label="vz50",fontsize=16,color="green",shape="box"];32[label="Nothing >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3];
33[label="Just vz70 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];33 -> 46[label="",style="solid", color="black", weight=3];
34[label="AProVE_IO vz4",fontsize=16,color="green",shape="box"];41[label="vz4",fontsize=16,color="green",shape="box"];42[label="vz50",fontsize=16,color="green",shape="box"];37[label="primbindIO (IO vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];37 -> 47[label="",style="solid", color="black", weight=3];
38[label="primbindIO (AProVE_IO vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];38 -> 48[label="",style="solid", color="black", weight=3];
39[label="primbindIO (AProVE_Exception vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];39 -> 49[label="",style="solid", color="black", weight=3];
40[label="primbindIO (AProVE_Error vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];40 -> 50[label="",style="solid", color="black", weight=3];
43 -> 61[label="",style="dashed", color="red", weight=0];
43[label="Monad.foldM0 vz3 vz51 vz60 ++ (vz61 >>= Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];43 -> 62[label="",style="dashed", color="magenta", weight=3];
43 -> 63[label="",style="dashed", color="magenta", weight=3];
44[label="[]",fontsize=16,color="green",shape="box"];45[label="Nothing",fontsize=16,color="green",shape="box"];46[label="Monad.foldM0 vz3 vz51 vz70",fontsize=16,color="black",shape="box"];46 -> 53[label="",style="solid", color="black", weight=3];
47[label="error []",fontsize=16,color="red",shape="box"];48[label="Monad.foldM0 vz3 vz51 vz80",fontsize=16,color="black",shape="box"];48 -> 54[label="",style="solid", color="black", weight=3];
49[label="AProVE_Exception vz80",fontsize=16,color="green",shape="box"];50[label="AProVE_Error vz80",fontsize=16,color="green",shape="box"];62[label="Monad.foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3];
63 -> 17[label="",style="dashed", color="red", weight=0];
63[label="vz61 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];63 -> 67[label="",style="dashed", color="magenta", weight=3];
61[label="vz10 ++ vz9",fontsize=16,color="burlywood",shape="triangle"];92[label="vz10/vz100 : vz101",fontsize=10,color="white",style="solid",shape="box"];61 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 68[label="",style="solid", color="burlywood", weight=3];
93[label="vz10/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 69[label="",style="solid", color="burlywood", weight=3];
53 -> 5[label="",style="dashed", color="red", weight=0];
53[label="Monad.foldM vz3 vz70 vz51",fontsize=16,color="magenta"];53 -> 57[label="",style="dashed", color="magenta", weight=3];
53 -> 58[label="",style="dashed", color="magenta", weight=3];
54 -> 5[label="",style="dashed", color="red", weight=0];
54[label="Monad.foldM vz3 vz80 vz51",fontsize=16,color="magenta"];54 -> 59[label="",style="dashed", color="magenta", weight=3];
54 -> 60[label="",style="dashed", color="magenta", weight=3];
66 -> 5[label="",style="dashed", color="red", weight=0];
66[label="Monad.foldM vz3 vz60 vz51",fontsize=16,color="magenta"];66 -> 70[label="",style="dashed", color="magenta", weight=3];
66 -> 71[label="",style="dashed", color="magenta", weight=3];
67[label="vz61",fontsize=16,color="green",shape="box"];68[label="(vz100 : vz101) ++ vz9",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="[] ++ vz9",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
57[label="vz51",fontsize=16,color="green",shape="box"];58[label="vz70",fontsize=16,color="green",shape="box"];59[label="vz51",fontsize=16,color="green",shape="box"];60[label="vz80",fontsize=16,color="green",shape="box"];70[label="vz51",fontsize=16,color="green",shape="box"];71[label="vz60",fontsize=16,color="green",shape="box"];72[label="vz100 : vz101 ++ vz9",fontsize=16,color="green",shape="box"];72 -> 74[label="",style="dashed", color="green", weight=3];
73[label="vz9",fontsize=16,color="green",shape="box"];74 -> 61[label="",style="dashed", color="red", weight=0];
74[label="vz101 ++ vz9",fontsize=16,color="magenta"];74 -> 75[label="",style="dashed", color="magenta", weight=3];
75[label="vz101",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs0(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_Maybe, h, ba)
   new_gtGtEs(vz3, vz51, h, ba) -> new_gtGtEs(vz3, vz51, h, ba)
   new_primbindIO(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_IO, h, ba)
   new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) -> new_gtGtEs(vz3, vz51, h, ba)
   new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_[], h, ba)
   new_foldM(vz3, :(vz50, vz51), ty_IO, h, ba) -> new_primbindIO(vz3, vz51, h, ba)
   new_foldM(vz3, :(vz50, vz51), ty_Maybe, h, ba) -> new_gtGtEs0(vz3, vz51, h, ba)

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

(10) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
----------------------------------------

(11)
Complex Obligation (AND)

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldM(vz3, :(vz50, vz51), ty_IO, h, ba) -> new_primbindIO(vz3, vz51, h, ba)
   new_primbindIO(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_IO, h, ba)

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

(13) 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_primbindIO(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_IO, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4, 4 >= 5


*new_foldM(vz3, :(vz50, vz51), ty_IO, h, ba) -> new_primbindIO(vz3, vz51, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 4 >= 3, 5 >= 4


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_[], h, ba)
   new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) -> new_gtGtEs(vz3, vz51, h, ba)
   new_gtGtEs(vz3, vz51, h, ba) -> new_gtGtEs(vz3, vz51, h, ba)

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

(16) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) -> new_gtGtEs(vz3, vz51, h, ba)


Used ordering: Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 1 + x_1 + x_2
   POL(new_foldM(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + 2*x_3 + x_4 + x_5
   POL(new_gtGtEs(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(ty_[]) = 0


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

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_[], h, ba)
   new_gtGtEs(vz3, vz51, h, ba) -> new_gtGtEs(vz3, vz51, h, ba)

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

(18) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(19)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(vz3, vz51, h, ba) -> new_gtGtEs(vz3, vz51, h, ba)

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

(20) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = new_gtGtEs(vz3, vz51, h, ba) evaluates to  t =new_gtGtEs(vz3, vz51, h, ba)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from new_gtGtEs(vz3, vz51, h, ba) to new_gtGtEs(vz3, vz51, h, ba).




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

(21)
NO

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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldM(vz3, :(vz50, vz51), ty_Maybe, h, ba) -> new_gtGtEs0(vz3, vz51, h, ba)
   new_gtGtEs0(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_Maybe, h, ba)

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

(23) 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_gtGtEs0(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_Maybe, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4, 4 >= 5


*new_foldM(vz3, :(vz50, vz51), ty_Maybe, h, ba) -> new_gtGtEs0(vz3, vz51, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 4 >= 3, 5 >= 4


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

(24)
YES

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

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(vz100, vz101), vz9, h) -> new_psPs(vz101, vz9, h)

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

(26) 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_psPs(:(vz100, vz101), vz9, h) -> new_psPs(vz101, vz9, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(27)
YES

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

(28) Narrow (COMPLETE)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.foldM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.foldM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.foldM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.foldM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];76[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 6[label="",style="solid", color="burlywood", weight=3];
77[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="Monad.foldM vz3 vz4 (vz50 : vz51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="Monad.foldM vz3 vz4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="blue",shape="box"];78[label=">>= :: (IO a) -> (a -> IO a) -> IO a",fontsize=10,color="white",style="solid",shape="box"];8 -> 78[label="",style="solid", color="blue", weight=9];
78 -> 10[label="",style="solid", color="blue", weight=3];
79[label=">>= :: ([] a) -> (a -> [] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];8 -> 79[label="",style="solid", color="blue", weight=9];
79 -> 11[label="",style="solid", color="blue", weight=3];
80[label=">>= :: (Maybe a) -> (a -> Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];8 -> 80[label="",style="solid", color="blue", weight=9];
80 -> 12[label="",style="solid", color="blue", weight=3];
9[label="return vz4",fontsize=16,color="blue",shape="box"];81[label="return :: a -> IO a",fontsize=10,color="white",style="solid",shape="box"];9 -> 81[label="",style="solid", color="blue", weight=9];
81 -> 13[label="",style="solid", color="blue", weight=3];
82[label="return :: a -> [] a",fontsize=10,color="white",style="solid",shape="box"];9 -> 82[label="",style="solid", color="blue", weight=9];
82 -> 14[label="",style="solid", color="blue", weight=3];
83[label="return :: a -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];9 -> 83[label="",style="solid", color="blue", weight=9];
83 -> 15[label="",style="solid", color="blue", weight=3];
10[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];10 -> 16[label="",style="solid", color="black", weight=3];
11 -> 17[label="",style="dashed", color="red", weight=0];
11[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];11 -> 18[label="",style="dashed", color="magenta", weight=3];
12 -> 19[label="",style="dashed", color="red", weight=0];
12[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];12 -> 20[label="",style="dashed", color="magenta", weight=3];
13[label="return vz4",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="return vz4",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="return vz4",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16 -> 24[label="",style="dashed", color="red", weight=0];
16[label="primbindIO (vz3 vz4 vz50) (Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];16 -> 25[label="",style="dashed", color="magenta", weight=3];
18[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];18 -> 26[label="",style="dashed", color="green", weight=3];
18 -> 27[label="",style="dashed", color="green", weight=3];
17[label="vz6 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];84[label="vz6/vz60 : vz61",fontsize=10,color="white",style="solid",shape="box"];17 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 28[label="",style="solid", color="burlywood", weight=3];
85[label="vz6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 29[label="",style="solid", color="burlywood", weight=3];
20[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];20 -> 30[label="",style="dashed", color="green", weight=3];
20 -> 31[label="",style="dashed", color="green", weight=3];
19[label="vz7 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];86[label="vz7/Nothing",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 32[label="",style="solid", color="burlywood", weight=3];
87[label="vz7/Just vz70",fontsize=10,color="white",style="solid",shape="box"];19 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 33[label="",style="solid", color="burlywood", weight=3];
21[label="primretIO vz4",fontsize=16,color="black",shape="box"];21 -> 34[label="",style="solid", color="black", weight=3];
22[label="vz4 : []",fontsize=16,color="green",shape="box"];23[label="Just vz4",fontsize=16,color="green",shape="box"];25[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];25 -> 41[label="",style="dashed", color="green", weight=3];
25 -> 42[label="",style="dashed", color="green", weight=3];
24[label="primbindIO vz8 (Monad.foldM0 vz3 vz51)",fontsize=16,color="burlywood",shape="triangle"];88[label="vz8/IO vz80",fontsize=10,color="white",style="solid",shape="box"];24 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 37[label="",style="solid", color="burlywood", weight=3];
89[label="vz8/AProVE_IO vz80",fontsize=10,color="white",style="solid",shape="box"];24 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 38[label="",style="solid", color="burlywood", weight=3];
90[label="vz8/AProVE_Exception vz80",fontsize=10,color="white",style="solid",shape="box"];24 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 39[label="",style="solid", color="burlywood", weight=3];
91[label="vz8/AProVE_Error vz80",fontsize=10,color="white",style="solid",shape="box"];24 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 40[label="",style="solid", color="burlywood", weight=3];
26[label="vz4",fontsize=16,color="green",shape="box"];27[label="vz50",fontsize=16,color="green",shape="box"];28[label="vz60 : vz61 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];28 -> 43[label="",style="solid", color="black", weight=3];
29[label="[] >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];29 -> 44[label="",style="solid", color="black", weight=3];
30[label="vz4",fontsize=16,color="green",shape="box"];31[label="vz50",fontsize=16,color="green",shape="box"];32[label="Nothing >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3];
33[label="Just vz70 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];33 -> 46[label="",style="solid", color="black", weight=3];
34[label="AProVE_IO vz4",fontsize=16,color="green",shape="box"];41[label="vz4",fontsize=16,color="green",shape="box"];42[label="vz50",fontsize=16,color="green",shape="box"];37[label="primbindIO (IO vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];37 -> 47[label="",style="solid", color="black", weight=3];
38[label="primbindIO (AProVE_IO vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];38 -> 48[label="",style="solid", color="black", weight=3];
39[label="primbindIO (AProVE_Exception vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];39 -> 49[label="",style="solid", color="black", weight=3];
40[label="primbindIO (AProVE_Error vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];40 -> 50[label="",style="solid", color="black", weight=3];
43 -> 61[label="",style="dashed", color="red", weight=0];
43[label="Monad.foldM0 vz3 vz51 vz60 ++ (vz61 >>= Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];43 -> 62[label="",style="dashed", color="magenta", weight=3];
43 -> 63[label="",style="dashed", color="magenta", weight=3];
44[label="[]",fontsize=16,color="green",shape="box"];45[label="Nothing",fontsize=16,color="green",shape="box"];46[label="Monad.foldM0 vz3 vz51 vz70",fontsize=16,color="black",shape="box"];46 -> 53[label="",style="solid", color="black", weight=3];
47[label="error []",fontsize=16,color="red",shape="box"];48[label="Monad.foldM0 vz3 vz51 vz80",fontsize=16,color="black",shape="box"];48 -> 54[label="",style="solid", color="black", weight=3];
49[label="AProVE_Exception vz80",fontsize=16,color="green",shape="box"];50[label="AProVE_Error vz80",fontsize=16,color="green",shape="box"];62[label="Monad.foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3];
63 -> 17[label="",style="dashed", color="red", weight=0];
63[label="vz61 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];63 -> 67[label="",style="dashed", color="magenta", weight=3];
61[label="vz10 ++ vz9",fontsize=16,color="burlywood",shape="triangle"];92[label="vz10/vz100 : vz101",fontsize=10,color="white",style="solid",shape="box"];61 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 68[label="",style="solid", color="burlywood", weight=3];
93[label="vz10/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 69[label="",style="solid", color="burlywood", weight=3];
53 -> 5[label="",style="dashed", color="red", weight=0];
53[label="Monad.foldM vz3 vz70 vz51",fontsize=16,color="magenta"];53 -> 57[label="",style="dashed", color="magenta", weight=3];
53 -> 58[label="",style="dashed", color="magenta", weight=3];
54 -> 5[label="",style="dashed", color="red", weight=0];
54[label="Monad.foldM vz3 vz80 vz51",fontsize=16,color="magenta"];54 -> 59[label="",style="dashed", color="magenta", weight=3];
54 -> 60[label="",style="dashed", color="magenta", weight=3];
66 -> 5[label="",style="dashed", color="red", weight=0];
66[label="Monad.foldM vz3 vz60 vz51",fontsize=16,color="magenta"];66 -> 70[label="",style="dashed", color="magenta", weight=3];
66 -> 71[label="",style="dashed", color="magenta", weight=3];
67[label="vz61",fontsize=16,color="green",shape="box"];68[label="(vz100 : vz101) ++ vz9",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="[] ++ vz9",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
57[label="vz51",fontsize=16,color="green",shape="box"];58[label="vz70",fontsize=16,color="green",shape="box"];59[label="vz51",fontsize=16,color="green",shape="box"];60[label="vz80",fontsize=16,color="green",shape="box"];70[label="vz51",fontsize=16,color="green",shape="box"];71[label="vz60",fontsize=16,color="green",shape="box"];72[label="vz100 : vz101 ++ vz9",fontsize=16,color="green",shape="box"];72 -> 74[label="",style="dashed", color="green", weight=3];
73[label="vz9",fontsize=16,color="green",shape="box"];74 -> 61[label="",style="dashed", color="red", weight=0];
74[label="vz101 ++ vz9",fontsize=16,color="magenta"];74 -> 75[label="",style="dashed", color="magenta", weight=3];
75[label="vz101",fontsize=16,color="green",shape="box"];}

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

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primbindIO(AProVE_IO(vz80), vz3, vz51, h, ba, []) -> new_foldM(vz3, vz80, vz51, ty_IO, h, ba, [])
   new_gtGtEs0(Just(vz70), vz3, vz51, h, ba, []) -> new_foldM(vz3, vz70, vz51, ty_Maybe, h, ba, [])

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

(30) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(31)
TRUE