/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 2 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES
    (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) 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;
liftM3 :: Monad a => (b  ->  c  ->  e  ->  d)  ->  a b  ->  a c  ->  a e  ->  a d;
liftM3 f m1 m2 m3 = m1 >>= (\x1 ->m2 >>= (\x2 ->m3 >>= (\x3 ->return (f x1 x2 x3))));

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\x3->return (f x1 x2 x3)"
is transformed to
"liftM30 f x1 x2 x3 = return (f x1 x2 x3);
"
The following Lambda expression
"\x2->m3 >>= liftM30 f x1 x2"
is transformed to
"liftM31 m3 f x1 x2 = m3 >>= liftM30 f x1 x2;
"
The following Lambda expression
"\x1->m2 >>= liftM31 m3 f x1"
is transformed to
"liftM32 m2 m3 f x1 = m2 >>= liftM31 m3 f x1;
"

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

(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;
liftM3 :: Monad a => (e  ->  b  ->  d  ->  c)  ->  a e  ->  a b  ->  a d  ->  a c;
liftM3 f m1 m2 m3 = m1 >>= liftM32 m2 m3 f;

liftM30 f x1 x2 x3 = return (f x1 x2 x3);

liftM31 m3 f x1 x2 = m3 >>= liftM30 f x1 x2;

liftM32 m2 m3 f x1 = m2 >>= liftM31 m3 f x1;

}

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

(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;
liftM3 :: Monad e => (c  ->  b  ->  a  ->  d)  ->  e c  ->  e b  ->  e a  ->  e d;
liftM3 f m1 m2 m3 = m1 >>= liftM32 m2 m3 f;

liftM30 f x1 x2 x3 = return (f x1 x2 x3);

liftM31 m3 f x1 x2 = m3 >>= liftM30 f x1 x2;

liftM32 m2 m3 f x1 = m2 >>= liftM31 m3 f x1;

}

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

(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;
liftM3 :: Monad e => (c  ->  a  ->  d  ->  b)  ->  e c  ->  e a  ->  e d  ->  e b;
liftM3 f m1 m2 m3 = m1 >>= liftM32 m2 m3 f;

liftM30 f x1 x2 x3 = return (f x1 x2 x3);

liftM31 m3 f x1 x2 = m3 >>= liftM30 f x1 x2;

liftM32 m2 m3 f x1 = m2 >>= liftM31 m3 f x1;

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.liftM3",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.liftM3 vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.liftM3 vy3 vy4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="Monad.liftM3 vy3 vy4 vy5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="Monad.liftM3 vy3 vy4 vy5 vy6",fontsize=16,color="black",shape="triangle"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="vy4 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="blue",shape="box"];126[label=">>= :: (IO a) -> (a -> IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];7 -> 126[label="",style="solid", color="blue", weight=9];
126 -> 8[label="",style="solid", color="blue", weight=3];
127[label=">>= :: ([] a) -> (a -> [] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];7 -> 127[label="",style="solid", color="blue", weight=9];
127 -> 9[label="",style="solid", color="blue", weight=3];
128[label=">>= :: (Maybe a) -> (a -> Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];7 -> 128[label="",style="solid", color="blue", weight=9];
128 -> 10[label="",style="solid", color="blue", weight=3];
8[label="vy4 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="vy4 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="burlywood",shape="triangle"];129[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];9 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 12[label="",style="solid", color="burlywood", weight=3];
130[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 13[label="",style="solid", color="burlywood", weight=3];
10[label="vy4 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="burlywood",shape="box"];131[label="vy4/Nothing",fontsize=10,color="white",style="solid",shape="box"];10 -> 131[label="",style="solid", color="burlywood", weight=9];
131 -> 14[label="",style="solid", color="burlywood", weight=3];
132[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];10 -> 132[label="",style="solid", color="burlywood", weight=9];
132 -> 15[label="",style="solid", color="burlywood", weight=3];
11[label="primbindIO vy4 (Monad.liftM32 vy5 vy6 vy3)",fontsize=16,color="burlywood",shape="box"];133[label="vy4/IO vy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 133[label="",style="solid", color="burlywood", weight=9];
133 -> 16[label="",style="solid", color="burlywood", weight=3];
134[label="vy4/AProVE_IO vy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 134[label="",style="solid", color="burlywood", weight=9];
134 -> 17[label="",style="solid", color="burlywood", weight=3];
135[label="vy4/AProVE_Exception vy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 135[label="",style="solid", color="burlywood", weight=9];
135 -> 18[label="",style="solid", color="burlywood", weight=3];
136[label="vy4/AProVE_Error vy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 136[label="",style="solid", color="burlywood", weight=9];
136 -> 19[label="",style="solid", color="burlywood", weight=3];
12[label="vy40 : vy41 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
13[label="[] >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="Nothing >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="Just vy40 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
16[label="primbindIO (IO vy40) (Monad.liftM32 vy5 vy6 vy3)",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
17[label="primbindIO (AProVE_IO vy40) (Monad.liftM32 vy5 vy6 vy3)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
18[label="primbindIO (AProVE_Exception vy40) (Monad.liftM32 vy5 vy6 vy3)",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
19[label="primbindIO (AProVE_Error vy40) (Monad.liftM32 vy5 vy6 vy3)",fontsize=16,color="black",shape="box"];19 -> 27[label="",style="solid", color="black", weight=3];
20 -> 70[label="",style="dashed", color="red", weight=0];
20[label="Monad.liftM32 vy5 vy6 vy3 vy40 ++ (vy41 >>= Monad.liftM32 vy5 vy6 vy3)",fontsize=16,color="magenta"];20 -> 71[label="",style="dashed", color="magenta", weight=3];
20 -> 72[label="",style="dashed", color="magenta", weight=3];
21[label="[]",fontsize=16,color="green",shape="box"];22[label="Nothing",fontsize=16,color="green",shape="box"];23[label="Monad.liftM32 vy5 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];23 -> 30[label="",style="solid", color="black", weight=3];
24[label="error []",fontsize=16,color="red",shape="box"];25[label="Monad.liftM32 vy5 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];25 -> 31[label="",style="solid", color="black", weight=3];
26[label="AProVE_Exception vy40",fontsize=16,color="green",shape="box"];27[label="AProVE_Error vy40",fontsize=16,color="green",shape="box"];71 -> 9[label="",style="dashed", color="red", weight=0];
71[label="vy41 >>= Monad.liftM32 vy5 vy6 vy3",fontsize=16,color="magenta"];71 -> 81[label="",style="dashed", color="magenta", weight=3];
72[label="Monad.liftM32 vy5 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];72 -> 82[label="",style="solid", color="black", weight=3];
70[label="vy8 ++ vy7",fontsize=16,color="burlywood",shape="triangle"];137[label="vy8/vy80 : vy81",fontsize=10,color="white",style="solid",shape="box"];70 -> 137[label="",style="solid", color="burlywood", weight=9];
137 -> 83[label="",style="solid", color="burlywood", weight=3];
138[label="vy8/[]",fontsize=10,color="white",style="solid",shape="box"];70 -> 138[label="",style="solid", color="burlywood", weight=9];
138 -> 84[label="",style="solid", color="burlywood", weight=3];
30[label="vy5 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="burlywood",shape="box"];139[label="vy5/Nothing",fontsize=10,color="white",style="solid",shape="box"];30 -> 139[label="",style="solid", color="burlywood", weight=9];
139 -> 34[label="",style="solid", color="burlywood", weight=3];
140[label="vy5/Just vy50",fontsize=10,color="white",style="solid",shape="box"];30 -> 140[label="",style="solid", color="burlywood", weight=9];
140 -> 35[label="",style="solid", color="burlywood", weight=3];
31[label="vy5 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3];
81[label="vy41",fontsize=16,color="green",shape="box"];82[label="vy5 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="burlywood",shape="triangle"];141[label="vy5/vy50 : vy51",fontsize=10,color="white",style="solid",shape="box"];82 -> 141[label="",style="solid", color="burlywood", weight=9];
141 -> 90[label="",style="solid", color="burlywood", weight=3];
142[label="vy5/[]",fontsize=10,color="white",style="solid",shape="box"];82 -> 142[label="",style="solid", color="burlywood", weight=9];
142 -> 91[label="",style="solid", color="burlywood", weight=3];
83[label="(vy80 : vy81) ++ vy7",fontsize=16,color="black",shape="box"];83 -> 92[label="",style="solid", color="black", weight=3];
84[label="[] ++ vy7",fontsize=16,color="black",shape="box"];84 -> 93[label="",style="solid", color="black", weight=3];
34[label="Nothing >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
35[label="Just vy50 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
36[label="primbindIO vy5 (Monad.liftM31 vy6 vy3 vy40)",fontsize=16,color="burlywood",shape="box"];143[label="vy5/IO vy50",fontsize=10,color="white",style="solid",shape="box"];36 -> 143[label="",style="solid", color="burlywood", weight=9];
143 -> 41[label="",style="solid", color="burlywood", weight=3];
144[label="vy5/AProVE_IO vy50",fontsize=10,color="white",style="solid",shape="box"];36 -> 144[label="",style="solid", color="burlywood", weight=9];
144 -> 42[label="",style="solid", color="burlywood", weight=3];
145[label="vy5/AProVE_Exception vy50",fontsize=10,color="white",style="solid",shape="box"];36 -> 145[label="",style="solid", color="burlywood", weight=9];
145 -> 43[label="",style="solid", color="burlywood", weight=3];
146[label="vy5/AProVE_Error vy50",fontsize=10,color="white",style="solid",shape="box"];36 -> 146[label="",style="solid", color="burlywood", weight=9];
146 -> 44[label="",style="solid", color="burlywood", weight=3];
90[label="vy50 : vy51 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];90 -> 96[label="",style="solid", color="black", weight=3];
91[label="[] >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="black",shape="box"];91 -> 97[label="",style="solid", color="black", weight=3];
92[label="vy80 : vy81 ++ vy7",fontsize=16,color="green",shape="box"];92 -> 98[label="",style="dashed", color="green", weight=3];
93[label="vy7",fontsize=16,color="green",shape="box"];39[label="Nothing",fontsize=16,color="green",shape="box"];40[label="Monad.liftM31 vy6 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
41[label="primbindIO (IO vy50) (Monad.liftM31 vy6 vy3 vy40)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
42[label="primbindIO (AProVE_IO vy50) (Monad.liftM31 vy6 vy3 vy40)",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
43[label="primbindIO (AProVE_Exception vy50) (Monad.liftM31 vy6 vy3 vy40)",fontsize=16,color="black",shape="box"];43 -> 50[label="",style="solid", color="black", weight=3];
44[label="primbindIO (AProVE_Error vy50) (Monad.liftM31 vy6 vy3 vy40)",fontsize=16,color="black",shape="box"];44 -> 51[label="",style="solid", color="black", weight=3];
96 -> 70[label="",style="dashed", color="red", weight=0];
96[label="Monad.liftM31 vy6 vy3 vy40 vy50 ++ (vy51 >>= Monad.liftM31 vy6 vy3 vy40)",fontsize=16,color="magenta"];96 -> 103[label="",style="dashed", color="magenta", weight=3];
96 -> 104[label="",style="dashed", color="magenta", weight=3];
97[label="[]",fontsize=16,color="green",shape="box"];98 -> 70[label="",style="dashed", color="red", weight=0];
98[label="vy81 ++ vy7",fontsize=16,color="magenta"];98 -> 105[label="",style="dashed", color="magenta", weight=3];
47[label="vy6 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="burlywood",shape="box"];147[label="vy6/Nothing",fontsize=10,color="white",style="solid",shape="box"];47 -> 147[label="",style="solid", color="burlywood", weight=9];
147 -> 54[label="",style="solid", color="burlywood", weight=3];
148[label="vy6/Just vy60",fontsize=10,color="white",style="solid",shape="box"];47 -> 148[label="",style="solid", color="burlywood", weight=9];
148 -> 55[label="",style="solid", color="burlywood", weight=3];
48[label="error []",fontsize=16,color="red",shape="box"];49[label="Monad.liftM31 vy6 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];49 -> 56[label="",style="solid", color="black", weight=3];
50[label="AProVE_Exception vy50",fontsize=16,color="green",shape="box"];51[label="AProVE_Error vy50",fontsize=16,color="green",shape="box"];103 -> 82[label="",style="dashed", color="red", weight=0];
103[label="vy51 >>= Monad.liftM31 vy6 vy3 vy40",fontsize=16,color="magenta"];103 -> 107[label="",style="dashed", color="magenta", weight=3];
104[label="Monad.liftM31 vy6 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];104 -> 108[label="",style="solid", color="black", weight=3];
105[label="vy81",fontsize=16,color="green",shape="box"];54[label="Nothing >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];54 -> 59[label="",style="solid", color="black", weight=3];
55[label="Just vy60 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];55 -> 60[label="",style="solid", color="black", weight=3];
56[label="vy6 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];56 -> 61[label="",style="solid", color="black", weight=3];
107[label="vy51",fontsize=16,color="green",shape="box"];108[label="vy6 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="burlywood",shape="triangle"];149[label="vy6/vy60 : vy61",fontsize=10,color="white",style="solid",shape="box"];108 -> 149[label="",style="solid", color="burlywood", weight=9];
149 -> 110[label="",style="solid", color="burlywood", weight=3];
150[label="vy6/[]",fontsize=10,color="white",style="solid",shape="box"];108 -> 150[label="",style="solid", color="burlywood", weight=9];
150 -> 111[label="",style="solid", color="burlywood", weight=3];
59[label="Nothing",fontsize=16,color="green",shape="box"];60[label="Monad.liftM30 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];60 -> 64[label="",style="solid", color="black", weight=3];
61[label="primbindIO vy6 (Monad.liftM30 vy3 vy40 vy50)",fontsize=16,color="burlywood",shape="box"];151[label="vy6/IO vy60",fontsize=10,color="white",style="solid",shape="box"];61 -> 151[label="",style="solid", color="burlywood", weight=9];
151 -> 65[label="",style="solid", color="burlywood", weight=3];
152[label="vy6/AProVE_IO vy60",fontsize=10,color="white",style="solid",shape="box"];61 -> 152[label="",style="solid", color="burlywood", weight=9];
152 -> 66[label="",style="solid", color="burlywood", weight=3];
153[label="vy6/AProVE_Exception vy60",fontsize=10,color="white",style="solid",shape="box"];61 -> 153[label="",style="solid", color="burlywood", weight=9];
153 -> 67[label="",style="solid", color="burlywood", weight=3];
154[label="vy6/AProVE_Error vy60",fontsize=10,color="white",style="solid",shape="box"];61 -> 154[label="",style="solid", color="burlywood", weight=9];
154 -> 68[label="",style="solid", color="burlywood", weight=3];
110[label="vy60 : vy61 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];110 -> 115[label="",style="solid", color="black", weight=3];
111[label="[] >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="black",shape="box"];111 -> 116[label="",style="solid", color="black", weight=3];
64[label="return (vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];64 -> 85[label="",style="solid", color="black", weight=3];
65[label="primbindIO (IO vy60) (Monad.liftM30 vy3 vy40 vy50)",fontsize=16,color="black",shape="box"];65 -> 86[label="",style="solid", color="black", weight=3];
66[label="primbindIO (AProVE_IO vy60) (Monad.liftM30 vy3 vy40 vy50)",fontsize=16,color="black",shape="box"];66 -> 87[label="",style="solid", color="black", weight=3];
67[label="primbindIO (AProVE_Exception vy60) (Monad.liftM30 vy3 vy40 vy50)",fontsize=16,color="black",shape="box"];67 -> 88[label="",style="solid", color="black", weight=3];
68[label="primbindIO (AProVE_Error vy60) (Monad.liftM30 vy3 vy40 vy50)",fontsize=16,color="black",shape="box"];68 -> 89[label="",style="solid", color="black", weight=3];
115 -> 70[label="",style="dashed", color="red", weight=0];
115[label="Monad.liftM30 vy3 vy40 vy50 vy60 ++ (vy61 >>= Monad.liftM30 vy3 vy40 vy50)",fontsize=16,color="magenta"];115 -> 117[label="",style="dashed", color="magenta", weight=3];
115 -> 118[label="",style="dashed", color="magenta", weight=3];
116[label="[]",fontsize=16,color="green",shape="box"];85[label="Just (vy3 vy40 vy50 vy60)",fontsize=16,color="green",shape="box"];85 -> 94[label="",style="dashed", color="green", weight=3];
86[label="error []",fontsize=16,color="red",shape="box"];87[label="Monad.liftM30 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];87 -> 95[label="",style="solid", color="black", weight=3];
88[label="AProVE_Exception vy60",fontsize=16,color="green",shape="box"];89[label="AProVE_Error vy60",fontsize=16,color="green",shape="box"];117 -> 108[label="",style="dashed", color="red", weight=0];
117[label="vy61 >>= Monad.liftM30 vy3 vy40 vy50",fontsize=16,color="magenta"];117 -> 119[label="",style="dashed", color="magenta", weight=3];
118[label="Monad.liftM30 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];118 -> 120[label="",style="solid", color="black", weight=3];
94[label="vy3 vy40 vy50 vy60",fontsize=16,color="green",shape="box"];94 -> 99[label="",style="dashed", color="green", weight=3];
94 -> 100[label="",style="dashed", color="green", weight=3];
94 -> 101[label="",style="dashed", color="green", weight=3];
95[label="return (vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];95 -> 102[label="",style="solid", color="black", weight=3];
119[label="vy61",fontsize=16,color="green",shape="box"];120[label="return (vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];120 -> 121[label="",style="solid", color="black", weight=3];
99[label="vy40",fontsize=16,color="green",shape="box"];100[label="vy50",fontsize=16,color="green",shape="box"];101[label="vy60",fontsize=16,color="green",shape="box"];102[label="primretIO (vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];102 -> 106[label="",style="solid", color="black", weight=3];
121[label="vy3 vy40 vy50 vy60 : []",fontsize=16,color="green",shape="box"];121 -> 122[label="",style="dashed", color="green", weight=3];
106[label="AProVE_IO (vy3 vy40 vy50 vy60)",fontsize=16,color="green",shape="box"];106 -> 109[label="",style="dashed", color="green", weight=3];
122[label="vy3 vy40 vy50 vy60",fontsize=16,color="green",shape="box"];122 -> 123[label="",style="dashed", color="green", weight=3];
122 -> 124[label="",style="dashed", color="green", weight=3];
122 -> 125[label="",style="dashed", color="green", weight=3];
109[label="vy3 vy40 vy50 vy60",fontsize=16,color="green",shape="box"];109 -> 112[label="",style="dashed", color="green", weight=3];
109 -> 113[label="",style="dashed", color="green", weight=3];
109 -> 114[label="",style="dashed", color="green", weight=3];
123[label="vy40",fontsize=16,color="green",shape="box"];124[label="vy50",fontsize=16,color="green",shape="box"];125[label="vy60",fontsize=16,color="green",shape="box"];112[label="vy40",fontsize=16,color="green",shape="box"];113[label="vy50",fontsize=16,color="green",shape="box"];114[label="vy60",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

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

   new_gtGtEs(:(vy60, vy61), vy3, vy40, vy50, h, ba, bb, bc) -> new_gtGtEs(vy61, vy3, vy40, vy50, h, ba, bb, bc)

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

(10) 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_gtGtEs(:(vy60, vy61), vy3, vy40, vy50, h, ba, bb, bc) -> new_gtGtEs(vy61, vy3, vy40, vy50, h, ba, bb, bc)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8


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

(11)
YES

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

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

   new_gtGtEs1(:(vy40, vy41), vy5, vy6, vy3, h, ba, bb, bc) -> new_gtGtEs1(vy41, vy5, vy6, vy3, h, ba, bb, bc)

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_gtGtEs1(:(vy40, vy41), vy5, vy6, vy3, h, ba, bb, bc) -> new_gtGtEs1(vy41, vy5, vy6, vy3, h, ba, bb, bc)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8


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

(14)
YES

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

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

   new_gtGtEs0(:(vy50, vy51), vy6, vy3, vy40, h, ba, bb, bc) -> new_gtGtEs0(vy51, vy6, vy3, vy40, h, ba, bb, bc)

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

(16) 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(:(vy50, vy51), vy6, vy3, vy40, h, ba, bb, bc) -> new_gtGtEs0(vy51, vy6, vy3, vy40, h, ba, bb, bc)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8


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

(17)
YES

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(vy80, vy81), vy7, h) -> new_psPs(vy81, vy7, h)

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

(19) 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(:(vy80, vy81), vy7, h) -> new_psPs(vy81, vy7, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(20)
YES