/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) IFR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) BR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) COR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) LetRed [EQUIVALENT, 0 ms]
(10) HASKELL
(11) NumRed [SOUND, 0 ms]
(12) HASKELL
(13) Narrow [SOUND, 0 ms]
(14) AND
    (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES
    (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\(m,_)->m"
is transformed to
"m0 (m,_) = m;
"
The following Lambda expression
"\(_,n)->n"
is transformed to
"n0 (_,n) = n;
"

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(3) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if m == 0 then 0 else n + floatDigits x"
is transformed to
"exponent0 x True = 0;
exponent0 x False = n + floatDigits x;
"

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(5) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(6)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(7) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(8)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(9) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"exponent0 x (m == 0) where {
exponent0 x True = 0;
exponent0 x False = n + floatDigits x;
;
m  = m0 vu10;
;
m0 (m,vw) = m;
;
n  = n0 vu10;
;
n0 (vv,n) = n;
;
vu10  = decodeFloat x;
} 
"
are unpacked to the following functions on top level
"exponentN0 ww (vv,n) = n;
"
"exponentVu10 ww = decodeFloat ww;
"
"exponentM ww = exponentM0 ww (exponentVu10 ww);
"
"exponentM0 ww (m,vw) = m;
"
"exponentN ww = exponentN0 ww (exponentVu10 ww);
"
"exponentExponent0 ww x True = 0;
exponentExponent0 ww x False = exponentN ww + floatDigits x;
"

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

(10)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(11) NumRed (SOUND)
Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
----------------------------------------

(12)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
}

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

(13) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="exponent",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="exponent wx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="exponentExponent0 wx3 wx3 (exponentM wx3 == fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5 -> 10[label="",style="dashed", color="red", weight=0];
5[label="exponentExponent0 wx3 wx3 (exponentM0 wx3 (exponentVu10 wx3) == fromInt (Pos Zero))",fontsize=16,color="magenta"];5 -> 11[label="",style="dashed", color="magenta", weight=3];
11[label="exponentVu10 wx3",fontsize=16,color="black",shape="triangle"];11 -> 16[label="",style="solid", color="black", weight=3];
10[label="exponentExponent0 wx3 wx3 (exponentM0 wx3 wx4 == fromInt (Pos Zero))",fontsize=16,color="burlywood",shape="triangle"];94[label="wx4/(wx40,wx41)",fontsize=10,color="white",style="solid",shape="box"];10 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="decodeFloat wx3",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
17[label="exponentExponent0 wx3 wx3 (exponentM0 wx3 (wx40,wx41) == fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
19[label="primFloatDecode wx3",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="exponentExponent0 wx3 wx3 (wx40 == fromInt (Pos Zero))",fontsize=16,color="burlywood",shape="box"];95[label="wx40/Integer wx400",fontsize=10,color="white",style="solid",shape="box"];20 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 22[label="",style="solid", color="burlywood", weight=3];
21[label="terminator wx3",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="exponentExponent0 wx3 wx3 (Integer wx400 == fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
23[label="ter1m wx3",fontsize=16,color="green",shape="box"];23 -> 25[label="",style="dashed", color="green", weight=3];
24[label="exponentExponent0 wx3 wx3 (Integer wx400 == Integer (Pos Zero))",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
25[label="wx3",fontsize=16,color="green",shape="box"];26[label="exponentExponent0 wx3 wx3 (primEqInt wx400 (Pos Zero))",fontsize=16,color="burlywood",shape="box"];96[label="wx400/Pos wx4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 27[label="",style="solid", color="burlywood", weight=3];
97[label="wx400/Neg wx4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 28[label="",style="solid", color="burlywood", weight=3];
27[label="exponentExponent0 wx3 wx3 (primEqInt (Pos wx4000) (Pos Zero))",fontsize=16,color="burlywood",shape="box"];98[label="wx4000/Succ wx40000",fontsize=10,color="white",style="solid",shape="box"];27 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 29[label="",style="solid", color="burlywood", weight=3];
99[label="wx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 30[label="",style="solid", color="burlywood", weight=3];
28[label="exponentExponent0 wx3 wx3 (primEqInt (Neg wx4000) (Pos Zero))",fontsize=16,color="burlywood",shape="box"];100[label="wx4000/Succ wx40000",fontsize=10,color="white",style="solid",shape="box"];28 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 31[label="",style="solid", color="burlywood", weight=3];
101[label="wx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 32[label="",style="solid", color="burlywood", weight=3];
29[label="exponentExponent0 wx3 wx3 (primEqInt (Pos (Succ wx40000)) (Pos Zero))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="exponentExponent0 wx3 wx3 (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="exponentExponent0 wx3 wx3 (primEqInt (Neg (Succ wx40000)) (Pos Zero))",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="exponentExponent0 wx3 wx3 (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="exponentExponent0 wx3 wx3 False",fontsize=16,color="black",shape="triangle"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="exponentExponent0 wx3 wx3 True",fontsize=16,color="black",shape="triangle"];34 -> 38[label="",style="solid", color="black", weight=3];
35 -> 33[label="",style="dashed", color="red", weight=0];
35[label="exponentExponent0 wx3 wx3 False",fontsize=16,color="magenta"];36 -> 34[label="",style="dashed", color="red", weight=0];
36[label="exponentExponent0 wx3 wx3 True",fontsize=16,color="magenta"];37[label="exponentN wx3 + floatDigits wx3",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="Pos Zero",fontsize=16,color="green",shape="box"];39[label="primPlusInt (exponentN wx3) (floatDigits wx3)",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3];
40 -> 41[label="",style="dashed", color="red", weight=0];
40[label="primPlusInt (exponentN0 wx3 (exponentVu10 wx3)) (floatDigits wx3)",fontsize=16,color="magenta"];40 -> 42[label="",style="dashed", color="magenta", weight=3];
42 -> 11[label="",style="dashed", color="red", weight=0];
42[label="exponentVu10 wx3",fontsize=16,color="magenta"];41[label="primPlusInt (exponentN0 wx3 wx5) (floatDigits wx3)",fontsize=16,color="burlywood",shape="triangle"];102[label="wx5/(wx50,wx51)",fontsize=10,color="white",style="solid",shape="box"];41 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 43[label="",style="solid", color="burlywood", weight=3];
43[label="primPlusInt (exponentN0 wx3 (wx50,wx51)) (floatDigits wx3)",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
44[label="primPlusInt wx51 (floatDigits wx3)",fontsize=16,color="burlywood",shape="box"];103[label="wx51/Pos wx510",fontsize=10,color="white",style="solid",shape="box"];44 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 45[label="",style="solid", color="burlywood", weight=3];
104[label="wx51/Neg wx510",fontsize=10,color="white",style="solid",shape="box"];44 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 46[label="",style="solid", color="burlywood", weight=3];
45[label="primPlusInt (Pos wx510) (floatDigits wx3)",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
46[label="primPlusInt (Neg wx510) (floatDigits wx3)",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="primPlusInt (Pos wx510) primFloatDigits",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48[label="primPlusInt (Neg wx510) primFloatDigits",fontsize=16,color="black",shape="box"];48 -> 50[label="",style="solid", color="black", weight=3];
49 -> 51[label="",style="dashed", color="red", weight=0];
49[label="primPlusInt (Pos wx510) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))",fontsize=16,color="magenta"];49 -> 52[label="",style="dashed", color="magenta", weight=3];
49 -> 53[label="",style="dashed", color="magenta", weight=3];
50 -> 54[label="",style="dashed", color="red", weight=0];
50[label="primPlusInt (Neg wx510) (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))",fontsize=16,color="magenta"];50 -> 55[label="",style="dashed", color="magenta", weight=3];
50 -> 56[label="",style="dashed", color="magenta", weight=3];
52[label="wx510",fontsize=16,color="green",shape="box"];53[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))",fontsize=16,color="green",shape="box"];51[label="primPlusInt (Pos wx7) (Pos (Succ wx8))",fontsize=16,color="black",shape="triangle"];51 -> 57[label="",style="solid", color="black", weight=3];
55[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))",fontsize=16,color="green",shape="box"];56[label="wx510",fontsize=16,color="green",shape="box"];54[label="primPlusInt (Neg wx10) (Pos (Succ wx11))",fontsize=16,color="black",shape="triangle"];54 -> 58[label="",style="solid", color="black", weight=3];
57[label="Pos (primPlusNat wx7 (Succ wx8))",fontsize=16,color="green",shape="box"];57 -> 59[label="",style="dashed", color="green", weight=3];
58[label="primMinusNat (Succ wx11) wx10",fontsize=16,color="burlywood",shape="box"];105[label="wx10/Succ wx100",fontsize=10,color="white",style="solid",shape="box"];58 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 60[label="",style="solid", color="burlywood", weight=3];
106[label="wx10/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 61[label="",style="solid", color="burlywood", weight=3];
59[label="primPlusNat wx7 (Succ wx8)",fontsize=16,color="burlywood",shape="box"];107[label="wx7/Succ wx70",fontsize=10,color="white",style="solid",shape="box"];59 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 62[label="",style="solid", color="burlywood", weight=3];
108[label="wx7/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 63[label="",style="solid", color="burlywood", weight=3];
60[label="primMinusNat (Succ wx11) (Succ wx100)",fontsize=16,color="black",shape="box"];60 -> 64[label="",style="solid", color="black", weight=3];
61[label="primMinusNat (Succ wx11) Zero",fontsize=16,color="black",shape="box"];61 -> 65[label="",style="solid", color="black", weight=3];
62[label="primPlusNat (Succ wx70) (Succ wx8)",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3];
63[label="primPlusNat Zero (Succ wx8)",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
64[label="primMinusNat wx11 wx100",fontsize=16,color="burlywood",shape="triangle"];109[label="wx11/Succ wx110",fontsize=10,color="white",style="solid",shape="box"];64 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 68[label="",style="solid", color="burlywood", weight=3];
110[label="wx11/Zero",fontsize=10,color="white",style="solid",shape="box"];64 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 69[label="",style="solid", color="burlywood", weight=3];
65[label="Pos (Succ wx11)",fontsize=16,color="green",shape="box"];66[label="Succ (Succ (primPlusNat wx70 wx8))",fontsize=16,color="green",shape="box"];66 -> 70[label="",style="dashed", color="green", weight=3];
67[label="Succ wx8",fontsize=16,color="green",shape="box"];68[label="primMinusNat (Succ wx110) wx100",fontsize=16,color="burlywood",shape="box"];111[label="wx100/Succ wx1000",fontsize=10,color="white",style="solid",shape="box"];68 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 71[label="",style="solid", color="burlywood", weight=3];
112[label="wx100/Zero",fontsize=10,color="white",style="solid",shape="box"];68 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 72[label="",style="solid", color="burlywood", weight=3];
69[label="primMinusNat Zero wx100",fontsize=16,color="burlywood",shape="box"];113[label="wx100/Succ wx1000",fontsize=10,color="white",style="solid",shape="box"];69 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 73[label="",style="solid", color="burlywood", weight=3];
114[label="wx100/Zero",fontsize=10,color="white",style="solid",shape="box"];69 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 74[label="",style="solid", color="burlywood", weight=3];
70[label="primPlusNat wx70 wx8",fontsize=16,color="burlywood",shape="triangle"];115[label="wx70/Succ wx700",fontsize=10,color="white",style="solid",shape="box"];70 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 75[label="",style="solid", color="burlywood", weight=3];
116[label="wx70/Zero",fontsize=10,color="white",style="solid",shape="box"];70 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 76[label="",style="solid", color="burlywood", weight=3];
71[label="primMinusNat (Succ wx110) (Succ wx1000)",fontsize=16,color="black",shape="box"];71 -> 77[label="",style="solid", color="black", weight=3];
72[label="primMinusNat (Succ wx110) Zero",fontsize=16,color="black",shape="box"];72 -> 78[label="",style="solid", color="black", weight=3];
73[label="primMinusNat Zero (Succ wx1000)",fontsize=16,color="black",shape="box"];73 -> 79[label="",style="solid", color="black", weight=3];
74[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];74 -> 80[label="",style="solid", color="black", weight=3];
75[label="primPlusNat (Succ wx700) wx8",fontsize=16,color="burlywood",shape="box"];117[label="wx8/Succ wx80",fontsize=10,color="white",style="solid",shape="box"];75 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 81[label="",style="solid", color="burlywood", weight=3];
118[label="wx8/Zero",fontsize=10,color="white",style="solid",shape="box"];75 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 82[label="",style="solid", color="burlywood", weight=3];
76[label="primPlusNat Zero wx8",fontsize=16,color="burlywood",shape="box"];119[label="wx8/Succ wx80",fontsize=10,color="white",style="solid",shape="box"];76 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 83[label="",style="solid", color="burlywood", weight=3];
120[label="wx8/Zero",fontsize=10,color="white",style="solid",shape="box"];76 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 84[label="",style="solid", color="burlywood", weight=3];
77 -> 64[label="",style="dashed", color="red", weight=0];
77[label="primMinusNat wx110 wx1000",fontsize=16,color="magenta"];77 -> 85[label="",style="dashed", color="magenta", weight=3];
77 -> 86[label="",style="dashed", color="magenta", weight=3];
78[label="Pos (Succ wx110)",fontsize=16,color="green",shape="box"];79[label="Neg (Succ wx1000)",fontsize=16,color="green",shape="box"];80[label="Pos Zero",fontsize=16,color="green",shape="box"];81[label="primPlusNat (Succ wx700) (Succ wx80)",fontsize=16,color="black",shape="box"];81 -> 87[label="",style="solid", color="black", weight=3];
82[label="primPlusNat (Succ wx700) Zero",fontsize=16,color="black",shape="box"];82 -> 88[label="",style="solid", color="black", weight=3];
83[label="primPlusNat Zero (Succ wx80)",fontsize=16,color="black",shape="box"];83 -> 89[label="",style="solid", color="black", weight=3];
84[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];84 -> 90[label="",style="solid", color="black", weight=3];
85[label="wx110",fontsize=16,color="green",shape="box"];86[label="wx1000",fontsize=16,color="green",shape="box"];87[label="Succ (Succ (primPlusNat wx700 wx80))",fontsize=16,color="green",shape="box"];87 -> 91[label="",style="dashed", color="green", weight=3];
88[label="Succ wx700",fontsize=16,color="green",shape="box"];89[label="Succ wx80",fontsize=16,color="green",shape="box"];90[label="Zero",fontsize=16,color="green",shape="box"];91 -> 70[label="",style="dashed", color="red", weight=0];
91[label="primPlusNat wx700 wx80",fontsize=16,color="magenta"];91 -> 92[label="",style="dashed", color="magenta", weight=3];
91 -> 93[label="",style="dashed", color="magenta", weight=3];
92[label="wx700",fontsize=16,color="green",shape="box"];93[label="wx80",fontsize=16,color="green",shape="box"];}

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

(14)
Complex Obligation (AND)

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

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

   new_primMinusNat(Succ(wx110), Succ(wx1000)) -> new_primMinusNat(wx110, wx1000)

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_primMinusNat(Succ(wx110), Succ(wx1000)) -> new_primMinusNat(wx110, wx1000)
The graph contains the following edges 1 > 1, 2 > 2


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

(17)
YES

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

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

   new_primPlusNat(Succ(wx700), Succ(wx80)) -> new_primPlusNat(wx700, wx80)

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_primPlusNat(Succ(wx700), Succ(wx80)) -> new_primPlusNat(wx700, wx80)
The graph contains the following edges 1 > 1, 2 > 2


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

(20)
YES