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


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


YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) 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;
ap :: Monad b => b (a  ->  c)  ->  b a  ->  b c;
ap = liftM2 id;

liftM2 :: Monad d => (c  ->  b  ->  a)  ->  d c  ->  d b  ->  d a;
liftM2 f m1 m2 = m1 >>= (\x1 ->m2 >>= (\x2 ->return (f x1 x2)));

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\x2->return (f x1 x2)"
is transformed to
"liftM20 f x1 x2 = return (f x1 x2);
"
The following Lambda expression
"\x1->m2 >>= liftM20 f x1"
is transformed to
"liftM21 m2 f x1 = m2 >>= liftM20 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;
ap :: Monad b => b (c  ->  a)  ->  b c  ->  b a;
ap = liftM2 id;

liftM2 :: Monad c => (d  ->  b  ->  a)  ->  c d  ->  c b  ->  c a;
liftM2 f m1 m2 = m1 >>= liftM21 m2 f;

liftM20 f x1 x2 = return (f x1 x2);

liftM21 m2 f x1 = m2 >>= liftM20 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;
ap :: Monad c => c (b  ->  a)  ->  c b  ->  c a;
ap = liftM2 id;

liftM2 :: Monad c => (b  ->  a  ->  d)  ->  c b  ->  c a  ->  c d;
liftM2 f m1 m2 = m1 >>= liftM21 m2 f;

liftM20 f x1 x2 = return (f x1 x2);

liftM21 m2 f x1 = m2 >>= liftM20 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;
ap :: Monad a => a (b  ->  c)  ->  a b  ->  a c;
ap = liftM2 id;

liftM2 :: Monad d => (c  ->  a  ->  b)  ->  d c  ->  d a  ->  d b;
liftM2 f m1 m2 = m1 >>= liftM21 m2 f;

liftM20 f x1 x2 = return (f x1 x2);

liftM21 m2 f x1 = m2 >>= liftM20 f x1;

}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.ap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.ap vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.ap vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="Monad.liftM2 id vy3 vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="blue",shape="box"];94[label=">>= :: (Maybe (a -> b)) -> ((a -> b) -> Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];6 -> 94[label="",style="solid", color="blue", weight=9];
94 -> 7[label="",style="solid", color="blue", weight=3];
95[label=">>= :: (IO (a -> b)) -> ((a -> b) -> IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];6 -> 95[label="",style="solid", color="blue", weight=9];
95 -> 8[label="",style="solid", color="blue", weight=3];
96[label=">>= :: ([] (a -> b)) -> ((a -> b) -> [] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];6 -> 96[label="",style="solid", color="blue", weight=9];
96 -> 9[label="",style="solid", color="blue", weight=3];
7[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="box"];97[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];7 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 10[label="",style="solid", color="burlywood", weight=3];
98[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];7 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];8 -> 12[label="",style="solid", color="black", weight=3];
9[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="triangle"];99[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];9 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 13[label="",style="solid", color="burlywood", weight=3];
100[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 14[label="",style="solid", color="burlywood", weight=3];
10[label="Nothing >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3];
11[label="Just vy30 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3];
12[label="primbindIO vy3 (Monad.liftM21 vy4 id)",fontsize=16,color="burlywood",shape="box"];101[label="vy3/IO vy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 17[label="",style="solid", color="burlywood", weight=3];
102[label="vy3/AProVE_IO vy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 18[label="",style="solid", color="burlywood", weight=3];
103[label="vy3/AProVE_Exception vy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 19[label="",style="solid", color="burlywood", weight=3];
104[label="vy3/AProVE_Error vy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 20[label="",style="solid", color="burlywood", weight=3];
13[label="vy30 : vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
14[label="[] >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
15[label="Nothing",fontsize=16,color="green",shape="box"];16[label="Monad.liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];16 -> 23[label="",style="solid", color="black", weight=3];
17[label="primbindIO (IO vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];17 -> 24[label="",style="solid", color="black", weight=3];
18[label="primbindIO (AProVE_IO vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3];
19[label="primbindIO (AProVE_Exception vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];19 -> 26[label="",style="solid", color="black", weight=3];
20[label="primbindIO (AProVE_Error vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3];
21 -> 28[label="",style="dashed", color="red", weight=0];
21[label="Monad.liftM21 vy4 id vy30 ++ (vy31 >>= Monad.liftM21 vy4 id)",fontsize=16,color="magenta"];21 -> 29[label="",style="dashed", color="magenta", weight=3];
22[label="[]",fontsize=16,color="green",shape="box"];23[label="vy4 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="box"];105[label="vy4/Nothing",fontsize=10,color="white",style="solid",shape="box"];23 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 30[label="",style="solid", color="burlywood", weight=3];
106[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];23 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 31[label="",style="solid", color="burlywood", weight=3];
24[label="error []",fontsize=16,color="red",shape="box"];25[label="Monad.liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];25 -> 32[label="",style="solid", color="black", weight=3];
26[label="AProVE_Exception vy30",fontsize=16,color="green",shape="box"];27[label="AProVE_Error vy30",fontsize=16,color="green",shape="box"];29 -> 9[label="",style="dashed", color="red", weight=0];
29[label="vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="magenta"];29 -> 33[label="",style="dashed", color="magenta", weight=3];
28[label="Monad.liftM21 vy4 id vy30 ++ vy5",fontsize=16,color="black",shape="triangle"];28 -> 34[label="",style="solid", color="black", weight=3];
30[label="Nothing >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];30 -> 35[label="",style="solid", color="black", weight=3];
31[label="Just vy40 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3];
32[label="vy4 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];32 -> 37[label="",style="solid", color="black", weight=3];
33[label="vy31",fontsize=16,color="green",shape="box"];34[label="(vy4 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="burlywood",shape="box"];107[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];34 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 38[label="",style="solid", color="burlywood", weight=3];
108[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];34 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 39[label="",style="solid", color="burlywood", weight=3];
35[label="Nothing",fontsize=16,color="green",shape="box"];36[label="Monad.liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="primbindIO vy4 (Monad.liftM20 id vy30)",fontsize=16,color="burlywood",shape="box"];109[label="vy4/IO vy40",fontsize=10,color="white",style="solid",shape="box"];37 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 41[label="",style="solid", color="burlywood", weight=3];
110[label="vy4/AProVE_IO vy40",fontsize=10,color="white",style="solid",shape="box"];37 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 42[label="",style="solid", color="burlywood", weight=3];
111[label="vy4/AProVE_Exception vy40",fontsize=10,color="white",style="solid",shape="box"];37 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 43[label="",style="solid", color="burlywood", weight=3];
112[label="vy4/AProVE_Error vy40",fontsize=10,color="white",style="solid",shape="box"];37 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 44[label="",style="solid", color="burlywood", weight=3];
38[label="(vy40 : vy41 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39[label="([] >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
40[label="return (id vy30 vy40)",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
41[label="primbindIO (IO vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
42[label="primbindIO (AProVE_IO vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
43[label="primbindIO (AProVE_Exception vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];43 -> 50[label="",style="solid", color="black", weight=3];
44[label="primbindIO (AProVE_Error vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];44 -> 51[label="",style="solid", color="black", weight=3];
45[label="(Monad.liftM20 id vy30 vy40 ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];45 -> 52[label="",style="solid", color="black", weight=3];
46[label="[] ++ vy5",fontsize=16,color="black",shape="triangle"];46 -> 53[label="",style="solid", color="black", weight=3];
47[label="Just (id vy30 vy40)",fontsize=16,color="green",shape="box"];47 -> 54[label="",style="dashed", color="green", weight=3];
48[label="error []",fontsize=16,color="red",shape="box"];49[label="Monad.liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];49 -> 55[label="",style="solid", color="black", weight=3];
50[label="AProVE_Exception vy40",fontsize=16,color="green",shape="box"];51[label="AProVE_Error vy40",fontsize=16,color="green",shape="box"];52[label="(return (id vy30 vy40) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
53[label="vy5",fontsize=16,color="green",shape="box"];54[label="id vy30 vy40",fontsize=16,color="black",shape="triangle"];54 -> 57[label="",style="solid", color="black", weight=3];
55 -> 58[label="",style="dashed", color="red", weight=0];
55[label="return (id vy30 vy40)",fontsize=16,color="magenta"];55 -> 59[label="",style="dashed", color="magenta", weight=3];
56 -> 60[label="",style="dashed", color="red", weight=0];
56[label="((id vy30 vy40 : []) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="magenta"];56 -> 61[label="",style="dashed", color="magenta", weight=3];
57[label="vy30 vy40",fontsize=16,color="green",shape="box"];57 -> 62[label="",style="dashed", color="green", weight=3];
59 -> 54[label="",style="dashed", color="red", weight=0];
59[label="id vy30 vy40",fontsize=16,color="magenta"];59 -> 63[label="",style="dashed", color="magenta", weight=3];
59 -> 64[label="",style="dashed", color="magenta", weight=3];
58[label="return vy6",fontsize=16,color="black",shape="triangle"];58 -> 65[label="",style="solid", color="black", weight=3];
61 -> 54[label="",style="dashed", color="red", weight=0];
61[label="id vy30 vy40",fontsize=16,color="magenta"];61 -> 66[label="",style="dashed", color="magenta", weight=3];
61 -> 67[label="",style="dashed", color="magenta", weight=3];
60[label="((vy7 : []) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="triangle"];60 -> 68[label="",style="solid", color="black", weight=3];
62[label="vy40",fontsize=16,color="green",shape="box"];63[label="vy30",fontsize=16,color="green",shape="box"];64[label="vy40",fontsize=16,color="green",shape="box"];65[label="primretIO vy6",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
66[label="vy30",fontsize=16,color="green",shape="box"];67[label="vy40",fontsize=16,color="green",shape="box"];68 -> 70[label="",style="dashed", color="red", weight=0];
68[label="(vy7 : [] ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="magenta"];68 -> 71[label="",style="dashed", color="magenta", weight=3];
69[label="AProVE_IO vy6",fontsize=16,color="green",shape="box"];71 -> 46[label="",style="dashed", color="red", weight=0];
71[label="[] ++ (vy41 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];71 -> 72[label="",style="dashed", color="magenta", weight=3];
70[label="(vy7 : vy8) ++ vy5",fontsize=16,color="black",shape="triangle"];70 -> 73[label="",style="solid", color="black", weight=3];
72[label="vy41 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="triangle"];113[label="vy41/vy410 : vy411",fontsize=10,color="white",style="solid",shape="box"];72 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 74[label="",style="solid", color="burlywood", weight=3];
114[label="vy41/[]",fontsize=10,color="white",style="solid",shape="box"];72 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 75[label="",style="solid", color="burlywood", weight=3];
73[label="vy7 : vy8 ++ vy5",fontsize=16,color="green",shape="box"];73 -> 76[label="",style="dashed", color="green", weight=3];
74[label="vy410 : vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];74 -> 77[label="",style="solid", color="black", weight=3];
75[label="[] >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];75 -> 78[label="",style="solid", color="black", weight=3];
76[label="vy8 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];115[label="vy8/vy80 : vy81",fontsize=10,color="white",style="solid",shape="box"];76 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 79[label="",style="solid", color="burlywood", weight=3];
116[label="vy8/[]",fontsize=10,color="white",style="solid",shape="box"];76 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 80[label="",style="solid", color="burlywood", weight=3];
77 -> 76[label="",style="dashed", color="red", weight=0];
77[label="Monad.liftM20 id vy30 vy410 ++ (vy411 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];77 -> 81[label="",style="dashed", color="magenta", weight=3];
77 -> 82[label="",style="dashed", color="magenta", weight=3];
78[label="[]",fontsize=16,color="green",shape="box"];79[label="(vy80 : vy81) ++ vy5",fontsize=16,color="black",shape="box"];79 -> 83[label="",style="solid", color="black", weight=3];
80[label="[] ++ vy5",fontsize=16,color="black",shape="box"];80 -> 84[label="",style="solid", color="black", weight=3];
81 -> 72[label="",style="dashed", color="red", weight=0];
81[label="vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="magenta"];81 -> 85[label="",style="dashed", color="magenta", weight=3];
82[label="Monad.liftM20 id vy30 vy410",fontsize=16,color="black",shape="box"];82 -> 86[label="",style="solid", color="black", weight=3];
83[label="vy80 : vy81 ++ vy5",fontsize=16,color="green",shape="box"];83 -> 87[label="",style="dashed", color="green", weight=3];
84[label="vy5",fontsize=16,color="green",shape="box"];85[label="vy411",fontsize=16,color="green",shape="box"];86 -> 88[label="",style="dashed", color="red", weight=0];
86[label="return (id vy30 vy410)",fontsize=16,color="magenta"];86 -> 89[label="",style="dashed", color="magenta", weight=3];
87 -> 76[label="",style="dashed", color="red", weight=0];
87[label="vy81 ++ vy5",fontsize=16,color="magenta"];87 -> 90[label="",style="dashed", color="magenta", weight=3];
89 -> 54[label="",style="dashed", color="red", weight=0];
89[label="id vy30 vy410",fontsize=16,color="magenta"];89 -> 91[label="",style="dashed", color="magenta", weight=3];
89 -> 92[label="",style="dashed", color="magenta", weight=3];
88[label="return vy9",fontsize=16,color="black",shape="triangle"];88 -> 93[label="",style="solid", color="black", weight=3];
90[label="vy81",fontsize=16,color="green",shape="box"];91[label="vy30",fontsize=16,color="green",shape="box"];92[label="vy410",fontsize=16,color="green",shape="box"];93[label="vy9 : []",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(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs0(vy31, vy4, h, ba)

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_gtGtEs0(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs0(vy31, vy4, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


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

(11)
YES

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

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

   new_gtGtEs(:(vy410, vy411), vy30, h, ba) -> new_gtGtEs(vy411, vy30, 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_gtGtEs(:(vy410, vy411), vy30, h, ba) -> new_gtGtEs(vy411, vy30, h, ba)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


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

(14)
YES

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

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

   new_psPs(:(vy80, vy81), vy5, h) -> new_psPs(vy81, vy5, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(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_psPs(:(vy80, vy81), vy5, h) -> new_psPs(vy81, vy5, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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(17)
YES