/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) 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 a => a (b  ->  c)  ->  a b  ->  a 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 c => c (b  ->  a)  ->  c b  ->  c 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 b => b (c  ->  a)  ->  b c  ->  b 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 b => b (a  ->  c)  ->  b a  ->  b 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"];93[label=">>= :: ([] (a -> b)) -> ((a -> b) -> [] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];6 -> 93[label="",style="solid", color="blue", weight=9];
93 -> 7[label="",style="solid", color="blue", weight=3];
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 -> 8[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 -> 9[label="",style="solid", color="blue", weight=3];
7[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="triangle"];96[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];7 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 10[label="",style="solid", color="burlywood", weight=3];
97[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 11[label="",style="solid", color="burlywood", weight=3];
8[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="box"];98[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 12[label="",style="solid", color="burlywood", weight=3];
99[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];8 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 13[label="",style="solid", color="burlywood", weight=3];
9[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];9 -> 14[label="",style="solid", color="black", weight=3];
10[label="vy30 : vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3];
11[label="[] >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3];
12[label="Nothing >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3];
13[label="Just vy30 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];13 -> 18[label="",style="solid", color="black", weight=3];
14[label="primbindIO vy3 (Monad.liftM21 vy4 id)",fontsize=16,color="burlywood",shape="box"];100[label="vy3/IO vy30",fontsize=10,color="white",style="solid",shape="box"];14 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 19[label="",style="solid", color="burlywood", weight=3];
101[label="vy3/AProVE_IO vy30",fontsize=10,color="white",style="solid",shape="box"];14 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 20[label="",style="solid", color="burlywood", weight=3];
102[label="vy3/AProVE_Exception vy30",fontsize=10,color="white",style="solid",shape="box"];14 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 21[label="",style="solid", color="burlywood", weight=3];
103[label="vy3/AProVE_Error vy30",fontsize=10,color="white",style="solid",shape="box"];14 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 22[label="",style="solid", color="burlywood", weight=3];
15 -> 23[label="",style="dashed", color="red", weight=0];
15[label="Monad.liftM21 vy4 id vy30 ++ (vy31 >>= Monad.liftM21 vy4 id)",fontsize=16,color="magenta"];15 -> 24[label="",style="dashed", color="magenta", weight=3];
16[label="[]",fontsize=16,color="green",shape="box"];17[label="Nothing",fontsize=16,color="green",shape="box"];18[label="Monad.liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3];
19[label="primbindIO (IO vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];19 -> 26[label="",style="solid", color="black", weight=3];
20[label="primbindIO (AProVE_IO vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3];
21[label="primbindIO (AProVE_Exception vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];21 -> 28[label="",style="solid", color="black", weight=3];
22[label="primbindIO (AProVE_Error vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];22 -> 29[label="",style="solid", color="black", weight=3];
24 -> 7[label="",style="dashed", color="red", weight=0];
24[label="vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="magenta"];24 -> 30[label="",style="dashed", color="magenta", weight=3];
23[label="Monad.liftM21 vy4 id vy30 ++ vy5",fontsize=16,color="black",shape="triangle"];23 -> 31[label="",style="solid", color="black", weight=3];
25[label="vy4 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="box"];104[label="vy4/Nothing",fontsize=10,color="white",style="solid",shape="box"];25 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 32[label="",style="solid", color="burlywood", weight=3];
105[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];25 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 33[label="",style="solid", color="burlywood", weight=3];
26[label="error []",fontsize=16,color="red",shape="box"];27[label="Monad.liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];27 -> 34[label="",style="solid", color="black", weight=3];
28[label="AProVE_Exception vy30",fontsize=16,color="green",shape="box"];29[label="AProVE_Error vy30",fontsize=16,color="green",shape="box"];30[label="vy31",fontsize=16,color="green",shape="box"];31[label="(vy4 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="burlywood",shape="box"];106[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];31 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 35[label="",style="solid", color="burlywood", weight=3];
107[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];31 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 36[label="",style="solid", color="burlywood", weight=3];
32[label="Nothing >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];32 -> 37[label="",style="solid", color="black", weight=3];
33[label="Just vy40 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];33 -> 38[label="",style="solid", color="black", weight=3];
34[label="vy4 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
35[label="(vy40 : vy41 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
36[label="([] >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
37[label="Nothing",fontsize=16,color="green",shape="box"];38[label="Monad.liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
39[label="primbindIO vy4 (Monad.liftM20 id vy30)",fontsize=16,color="burlywood",shape="box"];108[label="vy4/IO vy40",fontsize=10,color="white",style="solid",shape="box"];39 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 43[label="",style="solid", color="burlywood", weight=3];
109[label="vy4/AProVE_IO vy40",fontsize=10,color="white",style="solid",shape="box"];39 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 44[label="",style="solid", color="burlywood", weight=3];
110[label="vy4/AProVE_Exception vy40",fontsize=10,color="white",style="solid",shape="box"];39 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 45[label="",style="solid", color="burlywood", weight=3];
111[label="vy4/AProVE_Error vy40",fontsize=10,color="white",style="solid",shape="box"];39 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 46[label="",style="solid", color="burlywood", weight=3];
40[label="(Monad.liftM20 id vy30 vy40 ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
41[label="[] ++ vy5",fontsize=16,color="black",shape="triangle"];41 -> 48[label="",style="solid", color="black", weight=3];
42[label="return (id vy30 vy40)",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
43[label="primbindIO (IO vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];43 -> 50[label="",style="solid", color="black", weight=3];
44[label="primbindIO (AProVE_IO vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];44 -> 51[label="",style="solid", color="black", weight=3];
45[label="primbindIO (AProVE_Exception vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];45 -> 52[label="",style="solid", color="black", weight=3];
46[label="primbindIO (AProVE_Error vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];46 -> 53[label="",style="solid", color="black", weight=3];
47[label="(return (id vy30 vy40) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];47 -> 54[label="",style="solid", color="black", weight=3];
48[label="vy5",fontsize=16,color="green",shape="box"];49[label="Just (id vy30 vy40)",fontsize=16,color="green",shape="box"];49 -> 55[label="",style="dashed", color="green", weight=3];
50[label="error []",fontsize=16,color="red",shape="box"];51[label="Monad.liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3];
52[label="AProVE_Exception vy40",fontsize=16,color="green",shape="box"];53[label="AProVE_Error vy40",fontsize=16,color="green",shape="box"];54[label="((id vy30 vy40 : []) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
55[label="id vy30 vy40",fontsize=16,color="black",shape="triangle"];55 -> 58[label="",style="solid", color="black", weight=3];
56 -> 59[label="",style="dashed", color="red", weight=0];
56[label="return (id vy30 vy40)",fontsize=16,color="magenta"];56 -> 60[label="",style="dashed", color="magenta", weight=3];
57 -> 61[label="",style="dashed", color="red", weight=0];
57[label="(id vy30 vy40 : [] ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="magenta"];57 -> 62[label="",style="dashed", color="magenta", weight=3];
57 -> 63[label="",style="dashed", color="magenta", weight=3];
58[label="vy30 vy40",fontsize=16,color="green",shape="box"];58 -> 64[label="",style="dashed", color="green", weight=3];
60 -> 55[label="",style="dashed", color="red", weight=0];
60[label="id vy30 vy40",fontsize=16,color="magenta"];60 -> 65[label="",style="dashed", color="magenta", weight=3];
60 -> 66[label="",style="dashed", color="magenta", weight=3];
59[label="return vy6",fontsize=16,color="black",shape="triangle"];59 -> 67[label="",style="solid", color="black", weight=3];
62 -> 55[label="",style="dashed", color="red", weight=0];
62[label="id vy30 vy40",fontsize=16,color="magenta"];62 -> 68[label="",style="dashed", color="magenta", weight=3];
62 -> 69[label="",style="dashed", color="magenta", weight=3];
63 -> 41[label="",style="dashed", color="red", weight=0];
63[label="[] ++ (vy41 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];63 -> 70[label="",style="dashed", color="magenta", weight=3];
61[label="(vy8 : vy7) ++ vy5",fontsize=16,color="black",shape="triangle"];61 -> 71[label="",style="solid", color="black", weight=3];
64[label="vy40",fontsize=16,color="green",shape="box"];65[label="vy30",fontsize=16,color="green",shape="box"];66[label="vy40",fontsize=16,color="green",shape="box"];67[label="primretIO vy6",fontsize=16,color="black",shape="box"];67 -> 72[label="",style="solid", color="black", weight=3];
68[label="vy30",fontsize=16,color="green",shape="box"];69[label="vy40",fontsize=16,color="green",shape="box"];70[label="vy41 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="triangle"];112[label="vy41/vy410 : vy411",fontsize=10,color="white",style="solid",shape="box"];70 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 73[label="",style="solid", color="burlywood", weight=3];
113[label="vy41/[]",fontsize=10,color="white",style="solid",shape="box"];70 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 74[label="",style="solid", color="burlywood", weight=3];
71[label="vy8 : vy7 ++ vy5",fontsize=16,color="green",shape="box"];71 -> 75[label="",style="dashed", color="green", weight=3];
72[label="AProVE_IO vy6",fontsize=16,color="green",shape="box"];73[label="vy410 : vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];73 -> 76[label="",style="solid", color="black", weight=3];
74[label="[] >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];74 -> 77[label="",style="solid", color="black", weight=3];
75[label="vy7 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];114[label="vy7/vy70 : vy71",fontsize=10,color="white",style="solid",shape="box"];75 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 78[label="",style="solid", color="burlywood", weight=3];
115[label="vy7/[]",fontsize=10,color="white",style="solid",shape="box"];75 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 79[label="",style="solid", color="burlywood", weight=3];
76 -> 75[label="",style="dashed", color="red", weight=0];
76[label="Monad.liftM20 id vy30 vy410 ++ (vy411 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];76 -> 80[label="",style="dashed", color="magenta", weight=3];
76 -> 81[label="",style="dashed", color="magenta", weight=3];
77[label="[]",fontsize=16,color="green",shape="box"];78[label="(vy70 : vy71) ++ vy5",fontsize=16,color="black",shape="box"];78 -> 82[label="",style="solid", color="black", weight=3];
79[label="[] ++ vy5",fontsize=16,color="black",shape="box"];79 -> 83[label="",style="solid", color="black", weight=3];
80[label="Monad.liftM20 id vy30 vy410",fontsize=16,color="black",shape="box"];80 -> 84[label="",style="solid", color="black", weight=3];
81 -> 70[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="vy70 : vy71 ++ vy5",fontsize=16,color="green",shape="box"];82 -> 86[label="",style="dashed", color="green", weight=3];
83[label="vy5",fontsize=16,color="green",shape="box"];84 -> 87[label="",style="dashed", color="red", weight=0];
84[label="return (id vy30 vy410)",fontsize=16,color="magenta"];84 -> 88[label="",style="dashed", color="magenta", weight=3];
85[label="vy411",fontsize=16,color="green",shape="box"];86 -> 75[label="",style="dashed", color="red", weight=0];
86[label="vy71 ++ vy5",fontsize=16,color="magenta"];86 -> 89[label="",style="dashed", color="magenta", weight=3];
88 -> 55[label="",style="dashed", color="red", weight=0];
88[label="id vy30 vy410",fontsize=16,color="magenta"];88 -> 90[label="",style="dashed", color="magenta", weight=3];
88 -> 91[label="",style="dashed", color="magenta", weight=3];
87[label="return vy9",fontsize=16,color="black",shape="triangle"];87 -> 92[label="",style="solid", color="black", weight=3];
89[label="vy71",fontsize=16,color="green",shape="box"];90[label="vy30",fontsize=16,color="green",shape="box"];91[label="vy410",fontsize=16,color="green",shape="box"];92[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(:(vy70, vy71), vy5, h) -> new_psPs(vy71, 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(:(vy70, vy71), vy5, h) -> new_psPs(vy71, vy5, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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(17)
YES