/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) MRRProof [EQUIVALENT, 33 ms]
        (11) QDP
        (12) NonTerminationLoopProof [COMPLETE, 0 ms]
        (13) NO
    (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) YES
    (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) QDP
        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (25) YES
(26) Narrow [COMPLETE, 0 ms]
(27) AND
    (28) QDP
        (29) DependencyGraphProof [EQUIVALENT, 0 ms]
        (30) TRUE
    (31) QDP
        (32) DependencyGraphProof [EQUIVALENT, 0 ms]
        (33) 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 c => (b  ->  a  ->  c b)  ->  b  ->  [a]  ->  c b;
foldM _ a [] = return a;
foldM f a (x : xs) = f a x >>= (\fax ->foldM f fax xs);

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

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\_->q"
is transformed to
"gtGt0 q _ = q;
"
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 c => (a  ->  b  ->  c a)  ->  a  ->  [b]  ->  c a;
foldM _ a [] = return a;
foldM f a (x : xs) = f a x >>= foldM0 f xs;

foldM0 f xs fax = foldM f fax xs;

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

}

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

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

foldM0 f xs fax = foldM f fax xs;

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

}

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

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

foldM0 f xs fax = foldM f fax xs;

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

}

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

(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_ wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.foldM_ wu3 wu4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.foldM_ wu3 wu4 wu5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="Monad.foldM wu3 wu4 wu5 >> return ()",fontsize=16,color="blue",shape="box"];639[label=">> :: (Maybe a) -> (Maybe ()) -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];6 -> 639[label="",style="solid", color="blue", weight=9];
639 -> 7[label="",style="solid", color="blue", weight=3];
640[label=">> :: ([] a) -> ([] ()) -> [] ()",fontsize=10,color="white",style="solid",shape="box"];6 -> 640[label="",style="solid", color="blue", weight=9];
640 -> 8[label="",style="solid", color="blue", weight=3];
641[label=">> :: (IO a) -> (IO ()) -> IO ()",fontsize=10,color="white",style="solid",shape="box"];6 -> 641[label="",style="solid", color="blue", weight=9];
641 -> 9[label="",style="solid", color="blue", weight=3];
7[label="Monad.foldM wu3 wu4 wu5 >> return ()",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="Monad.foldM wu3 wu4 wu5 >> return ()",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="Monad.foldM wu3 wu4 wu5 >> return ()",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="Monad.foldM wu3 wu4 wu5 >>= gtGt0 (return ())",fontsize=16,color="burlywood",shape="triangle"];642[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];10 -> 642[label="",style="solid", color="burlywood", weight=9];
642 -> 13[label="",style="solid", color="burlywood", weight=3];
643[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 643[label="",style="solid", color="burlywood", weight=9];
643 -> 14[label="",style="solid", color="burlywood", weight=3];
11 -> 524[label="",style="dashed", color="red", weight=0];
11[label="Monad.foldM wu3 wu4 wu5 >>= gtGt0 (return ())",fontsize=16,color="magenta"];11 -> 525[label="",style="dashed", color="magenta", weight=3];
12[label="Monad.foldM wu3 wu4 wu5 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3];
13[label="Monad.foldM wu3 wu4 (wu50 : wu51) >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];13 -> 18[label="",style="solid", color="black", weight=3];
14[label="Monad.foldM wu3 wu4 [] >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];14 -> 19[label="",style="solid", color="black", weight=3];
525[label="Monad.foldM wu3 wu4 wu5",fontsize=16,color="burlywood",shape="triangle"];644[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];525 -> 644[label="",style="solid", color="burlywood", weight=9];
644 -> 579[label="",style="solid", color="burlywood", weight=3];
645[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];525 -> 645[label="",style="solid", color="burlywood", weight=9];
645 -> 580[label="",style="solid", color="burlywood", weight=3];
524[label="wu37 >>= gtGt0 (return ())",fontsize=16,color="burlywood",shape="triangle"];646[label="wu37/wu370 : wu371",fontsize=10,color="white",style="solid",shape="box"];524 -> 646[label="",style="solid", color="burlywood", weight=9];
646 -> 581[label="",style="solid", color="burlywood", weight=3];
647[label="wu37/[]",fontsize=10,color="white",style="solid",shape="box"];524 -> 647[label="",style="solid", color="burlywood", weight=9];
647 -> 582[label="",style="solid", color="burlywood", weight=3];
17[label="primbindIO (Monad.foldM wu3 wu4 wu5) (gtGt0 (return ()))",fontsize=16,color="burlywood",shape="triangle"];648[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];17 -> 648[label="",style="solid", color="burlywood", weight=9];
648 -> 22[label="",style="solid", color="burlywood", weight=3];
649[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 649[label="",style="solid", color="burlywood", weight=9];
649 -> 23[label="",style="solid", color="burlywood", weight=3];
18 -> 24[label="",style="dashed", color="red", weight=0];
18[label="wu3 wu4 wu50 >>= Monad.foldM0 wu3 wu51 >>= gtGt0 (return ())",fontsize=16,color="magenta"];18 -> 25[label="",style="dashed", color="magenta", weight=3];
19[label="return wu4 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];19 -> 26[label="",style="solid", color="black", weight=3];
579[label="Monad.foldM wu3 wu4 (wu50 : wu51)",fontsize=16,color="black",shape="box"];579 -> 583[label="",style="solid", color="black", weight=3];
580[label="Monad.foldM wu3 wu4 []",fontsize=16,color="black",shape="box"];580 -> 584[label="",style="solid", color="black", weight=3];
581[label="wu370 : wu371 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];581 -> 585[label="",style="solid", color="black", weight=3];
582[label="[] >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];582 -> 586[label="",style="solid", color="black", weight=3];
22[label="primbindIO (Monad.foldM wu3 wu4 (wu50 : wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
23[label="primbindIO (Monad.foldM wu3 wu4 []) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
25[label="wu3 wu4 wu50",fontsize=16,color="green",shape="box"];25 -> 32[label="",style="dashed", color="green", weight=3];
25 -> 33[label="",style="dashed", color="green", weight=3];
24[label="wu6 >>= Monad.foldM0 wu3 wu51 >>= gtGt0 (return ())",fontsize=16,color="burlywood",shape="triangle"];650[label="wu6/Nothing",fontsize=10,color="white",style="solid",shape="box"];24 -> 650[label="",style="solid", color="burlywood", weight=9];
650 -> 34[label="",style="solid", color="burlywood", weight=3];
651[label="wu6/Just wu60",fontsize=10,color="white",style="solid",shape="box"];24 -> 651[label="",style="solid", color="burlywood", weight=9];
651 -> 35[label="",style="solid", color="burlywood", weight=3];
26[label="Just wu4 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];26 -> 36[label="",style="solid", color="black", weight=3];
583 -> 587[label="",style="dashed", color="red", weight=0];
583[label="wu3 wu4 wu50 >>= Monad.foldM0 wu3 wu51",fontsize=16,color="magenta"];583 -> 588[label="",style="dashed", color="magenta", weight=3];
584[label="return wu4",fontsize=16,color="black",shape="triangle"];584 -> 589[label="",style="solid", color="black", weight=3];
585 -> 611[label="",style="dashed", color="red", weight=0];
585[label="gtGt0 (return ()) wu370 ++ (wu371 >>= gtGt0 (return ()))",fontsize=16,color="magenta"];585 -> 612[label="",style="dashed", color="magenta", weight=3];
585 -> 613[label="",style="dashed", color="magenta", weight=3];
586[label="[]",fontsize=16,color="green",shape="box"];30[label="primbindIO (wu3 wu4 wu50 >>= Monad.foldM0 wu3 wu51) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
31[label="primbindIO (return wu4) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
32[label="wu4",fontsize=16,color="green",shape="box"];33[label="wu50",fontsize=16,color="green",shape="box"];34[label="Nothing >>= Monad.foldM0 wu3 wu51 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];34 -> 46[label="",style="solid", color="black", weight=3];
35[label="Just wu60 >>= Monad.foldM0 wu3 wu51 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];35 -> 47[label="",style="solid", color="black", weight=3];
36[label="gtGt0 (return ()) wu4",fontsize=16,color="black",shape="box"];36 -> 48[label="",style="solid", color="black", weight=3];
588[label="wu3 wu4 wu50",fontsize=16,color="green",shape="box"];588 -> 593[label="",style="dashed", color="green", weight=3];
588 -> 594[label="",style="dashed", color="green", weight=3];
587[label="wu38 >>= Monad.foldM0 wu3 wu51",fontsize=16,color="burlywood",shape="triangle"];652[label="wu38/wu380 : wu381",fontsize=10,color="white",style="solid",shape="box"];587 -> 652[label="",style="solid", color="burlywood", weight=9];
652 -> 595[label="",style="solid", color="burlywood", weight=3];
653[label="wu38/[]",fontsize=10,color="white",style="solid",shape="box"];587 -> 653[label="",style="solid", color="burlywood", weight=9];
653 -> 596[label="",style="solid", color="burlywood", weight=3];
589[label="wu4 : []",fontsize=16,color="green",shape="box"];612 -> 524[label="",style="dashed", color="red", weight=0];
612[label="wu371 >>= gtGt0 (return ())",fontsize=16,color="magenta"];612 -> 624[label="",style="dashed", color="magenta", weight=3];
613 -> 625[label="",style="dashed", color="red", weight=0];
613[label="gtGt0 (return ()) wu370",fontsize=16,color="magenta"];613 -> 626[label="",style="dashed", color="magenta", weight=3];
611[label="wu43 ++ wu42",fontsize=16,color="burlywood",shape="triangle"];654[label="wu43/wu430 : wu431",fontsize=10,color="white",style="solid",shape="box"];611 -> 654[label="",style="solid", color="burlywood", weight=9];
654 -> 627[label="",style="solid", color="burlywood", weight=3];
655[label="wu43/[]",fontsize=10,color="white",style="solid",shape="box"];611 -> 655[label="",style="solid", color="burlywood", weight=9];
655 -> 628[label="",style="solid", color="burlywood", weight=3];
44 -> 52[label="",style="dashed", color="red", weight=0];
44[label="primbindIO (primbindIO (wu3 wu4 wu50) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="magenta"];44 -> 53[label="",style="dashed", color="magenta", weight=3];
45[label="primbindIO (primretIO wu4) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];45 -> 54[label="",style="solid", color="black", weight=3];
46[label="Nothing >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];46 -> 55[label="",style="solid", color="black", weight=3];
47[label="Monad.foldM0 wu3 wu51 wu60 >>= gtGt0 (return ())",fontsize=16,color="black",shape="box"];47 -> 56[label="",style="solid", color="black", weight=3];
48[label="return ()",fontsize=16,color="black",shape="box"];48 -> 57[label="",style="solid", color="black", weight=3];
593[label="wu4",fontsize=16,color="green",shape="box"];594[label="wu50",fontsize=16,color="green",shape="box"];595[label="wu380 : wu381 >>= Monad.foldM0 wu3 wu51",fontsize=16,color="black",shape="box"];595 -> 600[label="",style="solid", color="black", weight=3];
596[label="[] >>= Monad.foldM0 wu3 wu51",fontsize=16,color="black",shape="box"];596 -> 601[label="",style="solid", color="black", weight=3];
624[label="wu371",fontsize=16,color="green",shape="box"];626 -> 584[label="",style="dashed", color="red", weight=0];
626[label="return ()",fontsize=16,color="magenta"];626 -> 629[label="",style="dashed", color="magenta", weight=3];
625[label="gtGt0 wu44 wu370",fontsize=16,color="black",shape="triangle"];625 -> 630[label="",style="solid", color="black", weight=3];
627[label="(wu430 : wu431) ++ wu42",fontsize=16,color="black",shape="box"];627 -> 633[label="",style="solid", color="black", weight=3];
628[label="[] ++ wu42",fontsize=16,color="black",shape="box"];628 -> 634[label="",style="solid", color="black", weight=3];
53[label="wu3 wu4 wu50",fontsize=16,color="green",shape="box"];53 -> 62[label="",style="dashed", color="green", weight=3];
53 -> 63[label="",style="dashed", color="green", weight=3];
52[label="primbindIO (primbindIO wu8 (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="burlywood",shape="triangle"];656[label="wu8/IO wu80",fontsize=10,color="white",style="solid",shape="box"];52 -> 656[label="",style="solid", color="burlywood", weight=9];
656 -> 64[label="",style="solid", color="burlywood", weight=3];
657[label="wu8/AProVE_IO wu80",fontsize=10,color="white",style="solid",shape="box"];52 -> 657[label="",style="solid", color="burlywood", weight=9];
657 -> 65[label="",style="solid", color="burlywood", weight=3];
658[label="wu8/AProVE_Exception wu80",fontsize=10,color="white",style="solid",shape="box"];52 -> 658[label="",style="solid", color="burlywood", weight=9];
658 -> 66[label="",style="solid", color="burlywood", weight=3];
659[label="wu8/AProVE_Error wu80",fontsize=10,color="white",style="solid",shape="box"];52 -> 659[label="",style="solid", color="burlywood", weight=9];
659 -> 67[label="",style="solid", color="burlywood", weight=3];
54[label="primbindIO (AProVE_IO wu4) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];54 -> 68[label="",style="solid", color="black", weight=3];
55[label="Nothing",fontsize=16,color="green",shape="box"];56 -> 10[label="",style="dashed", color="red", weight=0];
56[label="Monad.foldM wu3 wu60 wu51 >>= gtGt0 (return ())",fontsize=16,color="magenta"];56 -> 69[label="",style="dashed", color="magenta", weight=3];
56 -> 70[label="",style="dashed", color="magenta", weight=3];
57[label="Just ()",fontsize=16,color="green",shape="box"];600 -> 611[label="",style="dashed", color="red", weight=0];
600[label="Monad.foldM0 wu3 wu51 wu380 ++ (wu381 >>= Monad.foldM0 wu3 wu51)",fontsize=16,color="magenta"];600 -> 618[label="",style="dashed", color="magenta", weight=3];
600 -> 619[label="",style="dashed", color="magenta", weight=3];
601[label="[]",fontsize=16,color="green",shape="box"];629[label="()",fontsize=16,color="green",shape="box"];630[label="wu44",fontsize=16,color="green",shape="box"];633[label="wu430 : wu431 ++ wu42",fontsize=16,color="green",shape="box"];633 -> 637[label="",style="dashed", color="green", weight=3];
634[label="wu42",fontsize=16,color="green",shape="box"];62[label="wu4",fontsize=16,color="green",shape="box"];63[label="wu50",fontsize=16,color="green",shape="box"];64[label="primbindIO (primbindIO (IO wu80) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3];
65[label="primbindIO (primbindIO (AProVE_IO wu80) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3];
66[label="primbindIO (primbindIO (AProVE_Exception wu80) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3];
67[label="primbindIO (primbindIO (AProVE_Error wu80) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
68[label="gtGt0 (return ()) wu4",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3];
69[label="wu51",fontsize=16,color="green",shape="box"];70[label="wu60",fontsize=16,color="green",shape="box"];618 -> 587[label="",style="dashed", color="red", weight=0];
618[label="wu381 >>= Monad.foldM0 wu3 wu51",fontsize=16,color="magenta"];618 -> 631[label="",style="dashed", color="magenta", weight=3];
619[label="Monad.foldM0 wu3 wu51 wu380",fontsize=16,color="black",shape="box"];619 -> 632[label="",style="solid", color="black", weight=3];
637 -> 611[label="",style="dashed", color="red", weight=0];
637[label="wu431 ++ wu42",fontsize=16,color="magenta"];637 -> 638[label="",style="dashed", color="magenta", weight=3];
74[label="error []",fontsize=16,color="red",shape="box"];75[label="primbindIO (Monad.foldM0 wu3 wu51 wu80) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];75 -> 82[label="",style="solid", color="black", weight=3];
76[label="primbindIO (AProVE_Exception wu80) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];76 -> 83[label="",style="solid", color="black", weight=3];
77[label="primbindIO (AProVE_Error wu80) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];77 -> 84[label="",style="solid", color="black", weight=3];
78[label="return ()",fontsize=16,color="black",shape="box"];78 -> 85[label="",style="solid", color="black", weight=3];
631[label="wu381",fontsize=16,color="green",shape="box"];632 -> 525[label="",style="dashed", color="red", weight=0];
632[label="Monad.foldM wu3 wu380 wu51",fontsize=16,color="magenta"];632 -> 635[label="",style="dashed", color="magenta", weight=3];
632 -> 636[label="",style="dashed", color="magenta", weight=3];
638[label="wu431",fontsize=16,color="green",shape="box"];82 -> 17[label="",style="dashed", color="red", weight=0];
82[label="primbindIO (Monad.foldM wu3 wu80 wu51) (gtGt0 (return ()))",fontsize=16,color="magenta"];82 -> 90[label="",style="dashed", color="magenta", weight=3];
82 -> 91[label="",style="dashed", color="magenta", weight=3];
83[label="AProVE_Exception wu80",fontsize=16,color="green",shape="box"];84[label="AProVE_Error wu80",fontsize=16,color="green",shape="box"];85[label="primretIO ()",fontsize=16,color="black",shape="box"];85 -> 92[label="",style="solid", color="black", weight=3];
635[label="wu51",fontsize=16,color="green",shape="box"];636[label="wu380",fontsize=16,color="green",shape="box"];90[label="wu51",fontsize=16,color="green",shape="box"];91[label="wu80",fontsize=16,color="green",shape="box"];92[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

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

   new_foldM(wu3, :(wu50, wu51), h, ba) -> new_gtGtEs(wu3, wu51, h, ba)
   new_gtGtEs(wu3, wu51, h, ba) -> new_gtGtEs(wu3, wu51, h, ba)
   new_gtGtEs(wu3, wu51, h, ba) -> new_foldM(wu3, wu51, h, ba)

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

(10) 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(wu3, :(wu50, wu51), h, ba) -> new_gtGtEs(wu3, wu51, h, ba)
   new_gtGtEs(wu3, wu51, h, ba) -> new_foldM(wu3, wu51, h, ba)


Used ordering: Polynomial interpretation [POLO]:

   POL(:(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(new_foldM(x_1, x_2, x_3, x_4)) = x_1 + 2*x_2 + x_3 + x_4
   POL(new_gtGtEs(x_1, x_2, x_3, x_4)) = 2 + x_1 + 2*x_2 + x_3 + x_4


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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs(wu3, wu51, h, ba) -> new_gtGtEs(wu3, wu51, h, ba)

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

(12) 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(wu3, wu51, h, ba) evaluates to  t =new_gtGtEs(wu3, wu51, 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(wu3, wu51, h, ba) to new_gtGtEs(wu3, wu51, h, ba).




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

(13)
NO

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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primbindIO(wu3, wu51, h, ba) -> new_primbindIO0(wu3, wu51, h, ba)
   new_primbindIO0(wu3, :(wu50, wu51), h, ba) -> new_primbindIO(wu3, wu51, h, ba)

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

(15) 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_primbindIO0(wu3, :(wu50, wu51), h, ba) -> new_primbindIO(wu3, wu51, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*new_primbindIO(wu3, wu51, h, ba) -> new_primbindIO0(wu3, wu51, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4


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

(16)
YES

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

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

   new_psPs(:(wu430, wu431), wu42, h) -> new_psPs(wu431, wu42, h)

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

(18) 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(:(wu430, wu431), wu42, h) -> new_psPs(wu431, wu42, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(19)
YES

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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs2(wu3, :(wu50, wu51), h, ba) -> new_gtGtEs1(wu3, wu51, h, ba)
   new_gtGtEs1(wu3, wu51, h, ba) -> new_gtGtEs2(wu3, wu51, h, ba)

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

(21) 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_gtGtEs1(wu3, wu51, h, ba) -> new_gtGtEs2(wu3, wu51, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4


*new_gtGtEs2(wu3, :(wu50, wu51), h, ba) -> new_gtGtEs1(wu3, wu51, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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

(22)
YES

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

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs0(:(wu370, wu371), h) -> new_gtGtEs0(wu371, h)

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

(24) 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(:(wu370, wu371), h) -> new_gtGtEs0(wu371, h)
The graph contains the following edges 1 > 1, 2 >= 2


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

(25)
YES

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

(26) Narrow (COMPLETE)
Haskell To QDPs

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67[label="primbindIO (primbindIO (AProVE_Error wu80) (Monad.foldM0 wu3 wu51)) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3];
68[label="gtGt0 (return ()) wu4",fontsize=16,color="black",shape="box"];68 -> 78[label="",style="solid", color="black", weight=3];
69[label="wu51",fontsize=16,color="green",shape="box"];70[label="wu60",fontsize=16,color="green",shape="box"];618 -> 587[label="",style="dashed", color="red", weight=0];
618[label="wu381 >>= Monad.foldM0 wu3 wu51",fontsize=16,color="magenta"];618 -> 631[label="",style="dashed", color="magenta", weight=3];
619[label="Monad.foldM0 wu3 wu51 wu380",fontsize=16,color="black",shape="box"];619 -> 632[label="",style="solid", color="black", weight=3];
637 -> 611[label="",style="dashed", color="red", weight=0];
637[label="wu431 ++ wu42",fontsize=16,color="magenta"];637 -> 638[label="",style="dashed", color="magenta", weight=3];
74[label="error []",fontsize=16,color="red",shape="box"];75[label="primbindIO (Monad.foldM0 wu3 wu51 wu80) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];75 -> 82[label="",style="solid", color="black", weight=3];
76[label="primbindIO (AProVE_Exception wu80) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];76 -> 83[label="",style="solid", color="black", weight=3];
77[label="primbindIO (AProVE_Error wu80) (gtGt0 (return ()))",fontsize=16,color="black",shape="box"];77 -> 84[label="",style="solid", color="black", weight=3];
78[label="return ()",fontsize=16,color="black",shape="box"];78 -> 85[label="",style="solid", color="black", weight=3];
631[label="wu381",fontsize=16,color="green",shape="box"];632 -> 525[label="",style="dashed", color="red", weight=0];
632[label="Monad.foldM wu3 wu380 wu51",fontsize=16,color="magenta"];632 -> 635[label="",style="dashed", color="magenta", weight=3];
632 -> 636[label="",style="dashed", color="magenta", weight=3];
638[label="wu431",fontsize=16,color="green",shape="box"];82 -> 17[label="",style="dashed", color="red", weight=0];
82[label="primbindIO (Monad.foldM wu3 wu80 wu51) (gtGt0 (return ()))",fontsize=16,color="magenta"];82 -> 90[label="",style="dashed", color="magenta", weight=3];
82 -> 91[label="",style="dashed", color="magenta", weight=3];
83[label="AProVE_Exception wu80",fontsize=16,color="green",shape="box"];84[label="AProVE_Error wu80",fontsize=16,color="green",shape="box"];85[label="primretIO ()",fontsize=16,color="black",shape="box"];85 -> 92[label="",style="solid", color="black", weight=3];
635[label="wu51",fontsize=16,color="green",shape="box"];636[label="wu380",fontsize=16,color="green",shape="box"];90[label="wu51",fontsize=16,color="green",shape="box"];91[label="wu80",fontsize=16,color="green",shape="box"];92[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];}

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

(27)
Complex Obligation (AND)

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

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_gtGtEs1(Just(wu60), wu3, wu51, h, ba, []) -> new_gtGtEs2(wu3, wu60, wu51, h, ba, [])

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

(29) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(30)
TRUE

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

(31)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primbindIO(AProVE_IO(wu80), wu3, wu51, h, ba, []) -> new_primbindIO0(wu3, wu80, wu51, h, ba, [])

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

(32) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(33)
TRUE