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okay  welcome everybody so we are um  here today lecture number 21 uh coming  into the home stretch of the course  uh i'd say probably  um this last quarter of the class of the  course is a bit more technical uh  perhaps so a little bit more abstract  some of the theorems are going to be  more difficult um so  i'll i'll try to work through them  slowly and answer your questions but uh  um  you know it's i think you can expect  them to find the material a bit more  challenging as we started uh  um  [Music]  uh  as we started um  um  you know with uh  this theorem last time uh  uh  uh  non-deterministic log space being closed  under complement so nl equals colonel  we kind of only got about part  maybe a third of the way through that so  i'm going to start over with that and  spend  kind of the first half of the lecture  today talking about that and then we're  going to talk about the  hierarchy theorems which are very  important um  uh aspect of the complexity landscape  basically they tell you  that if you allow  you know your favorite model let's say  turing machines to have more resources  then they can do more things  um but we'll get to that in due course  okay so let us  uh  go back  to oops reminder to myself  um  go back to  the immerman celeb cheney  uh which is that nl is  equal to co nl  um  so  as i mentioned these are going to be the  same slides as last time uh and i'll  just try to walk through them slowly  um i hope hope i hope you i hope you uh  get it but if you don't you know uh  ask me at the ta's questions  so we're gonna first i mean  the the the the i  the thing we're gonna show is that the  complement of path  is solvable in non-deterministic log  space we already know that path  is solvable in nl that's easy to do you  basically just  start at the start node  and you guess the sequence of nodes  storing only the current node in your  log space working memory  on the on your in your logs based work  tape  you guess the sequence of nodes  on the different branches of the  non-determinism and if you ever get to  the target node t then you can accept um  but how can a non-deterministic log  space machine know  or accept the complement of path so it  would have to accept when there's no  path  um and that uh is  a lot harder  but it's a big surprise to the  complexity community that it is it is  true  um  so as we uh discussed last time we're  going to talk about  computing functions with a  non-deterministic machine and that turns  out to be a convenient way of looking at  this  so we're going to have non-deterministic  machines  that have different branches of their  non-determinism you know on some input  and they're supposed to compute some  function value  you know remaining on the tape  but because of the non-determinism you  can't imagine that different branches  might have different function outputs  well that's not allowed  all branches must either report the  value of the function  um that we're trying to compute  or they can punt  basically they can reject  and say well you know uh  i i you know it's a basically i i don't  know  so all branches can must either report  the correct answer or they can say i  don't know and some branch must at least  one branch must report an answer  must report the answer  um for and that's what it means to be  computing a function  with a non-deterministic machine  and we're going to show that certain  functions can be computed with  non-deterministic log space machines  um  in particular this path function  which sort of  incorporates both the positive and  negative  of the pair both when there is a path  and when there is an is not a path  into into the function because the  function has to answer yes when there is  a path from s to t and no when there is  no path from s to t  okay so  if you could do this you're done  because  um you can make a non-deterministic you  could make an nl machine  so if you could compute the path  function you could make an nl machine  which would accept whenever the function  says no  um  and the other cases you know  and if  the um machine that's computing the  function  uh rejects you can then you'll reject as  well  but you accept if the function says no  and so therefore you're going to be  making a an nl machine which does the  complement of the path problem so it's  uh  you know  if you can compute the path function  that would be great  so that's what we would like to be able  to do  so as i mentioned we're going to have  two other values that that are going to  be relevant to computing the path  function which is what we're ultimately  going to do  and that's going to be the number of  nodes that you can reach from the start  uh  from the the start node  in your graph and the and then  for r is is the collection of nodes and  c is the number of reachable nodes  um  so shown on this picture and here i'm if  it's  helpful to you to see it in a more form  is  you  can think of r as a function of the  graph and the start node of course but  sometimes we'll just call it r you know  when it's clear which graph and start  node we're talking about  r is the set of reachable nodes so it's  the collection u  such that  the answer is  yes and  c  is the size of r  okay  so  the way we're going to start  is kind of an easy  theorem  though this is still going to be  relevant kind of at the end um but for  now it's really more a practice with the  concept that we have come up you know  this function concept that we've just  introduced so i want to say that  the path function  with an nl machine  then i can compute the count  uh  with an nl machine  okay so understand what computing the  path function means  that you have your nl machine  and  um  every branch has to either say i don't  know which is reject or it has to have  the answer which in it's going to be yes  if there is a path and know if there is  no path  and if i can do  be able to count the number of nodes  that do have a path  the number of notes for which the answer  is yes  and this  i think if you're comfortable with the  definition  this is  [Music]  more or less obvious  because  what you would do  is you would go through the nodes of g  one by one  and test uh  using your path function  uh what the answer is  yes or no  and every time it's a yes you add one to  the count until you've gone through all  of the nodes  and then um  you have your answer which is nodes  that's c  now if the machine  that's  trying to compute the path function on  the non-determinism rejects that that's  okay you'll  computing c that branch will reject also  but when you're um some branch  has to get the right answer on the on uh  you know for it has to get the right  answer  um and so then you get the  you know  you know what's happening with that  node and so you then um  can either increment the count or you  move on to the next node i think i'm  uh  trying to say  i'm not sure if i'm making any clear by  kind of repeating myself but okay here  so you're going to start out with the  you're given the  the graph and the start node  we're trying to compute this value c  which is the number rechargeable from  the start node  you start out with you you have a  counter which you're going to set  initially to zero and you go through  every node of the graph  and if uh  the path function computation says yes  you can reach you then you add one to  the count it says no you cannot reach  you then you just continue  um  and uh  maybe i should add another line here if  if the  you know if the if the the thing that's  computing  uh rejects then you also reject um  and then at the end you output  um  so what we're going to prove is the  other direction if i give you the count  then i can answer  the question for each node whether it's  reachable or not and this is the thing  because what it's saying that is i can  give you the count i'm done  if we can get that count that's going to  be enough  um  okay so  maybe even before the check-in maybe we  should just answer any questions because  you know if if you're  stuck here then you then you're doomed  um so i think it's you it makes sense to  try to understand what's going on at  this at this  um because i think the real the real  guts of this proof is coming on the next  slide um they're  kind of the main idea so i'm happy to  take  if there's any questions about this  i'll just wait for a second to see if  you're typing away  there well why don't we go to the  check-in maybe that'll help um  not a very difficult check-in  um  it'll come up  okay just a little practice with the  concept  so  i'm getting going to give you some graph  it has nine nodes  um and i want to know the value of the  count  um  so we'll we'll assume that s  the start node here is reachable from  itself  and now what's the value of c  okay  are we  good gonna shut this down give you  another two seconds please get your  answer in  okay  uh ready set end  yeah the right answer is in fact  uh e which is six there are six  reachable nodes in this graph and that's  what the value c is supposed to tell you  is how many nodes can i get to from s  okay and what i'm saying is that if i  can calculate that  in this sort of non-deterministic  function sense so if it's so some  branches can get that answer  um  then i can use that to  test for each node  whether it's reachable or not which is  kind of a little bit of a miracle  right that's kind of surprising just  knowing how many nodes are reachable  will allow me to test whether each  individual node is reachable  um because that's there's no obvious  reason why that would be  so there's uh going to be a procedure  for doing that which is on the next  slide and here it is  um  okay so this is the key idea that we're  going to repeat  um uh  later but  so  it's good to understand um  this is this is the slide you really  need to understand  uh  okay so i'm given the graph let's assume  the graph has m nodes  um  now as i said  okay so let's just say what what are we  doing here  given that count we can compute path  so we'll get the answer for every node  in the graph  if i just know  how many you know so i'll know i can get  the answer for which  whether a node is reachable or not  if i just have if i just know how many  reachable nodes there are  um so what i'm going to do is get get  that count of how many are reachable  now i'm going to go through  um  i'm going to because let's see what's  the idea here uh before we even jump  into the algorithm  the idea is  let's say i know how many nodes are  reachable like 100 nodes are reachable  now what i'm going to do what the  algorithm is going to do is find all  hundred reachable nodes  it's got one by one but it doesn't  matter sort of conceptually it's going  to find all reachable all reachable  nodes  and non-deterministically guessing them  so it's not sure in advance which ones  they are but it's going to guess  basically a hundred no it's going to  guess some of the nodes as being  reachable  confirm that the ones that guest are  reachable are reachable and then check  to see that that number equals 100.  on some branch of the non-determinism  you will guess right and you'll end up  with exactly the right set of 100  reachable nodes and then you'll see  is t  one of those reachable ones um in which  case you say yes or is t not one what is  not one of those hundred nodes  and then um you know the answer is no  because if you've guessed a hundred  nodes and you know they're all reachable  and you know there are exactly 100  reachable nodes then every other node is  not reachable  so that's kind of that's the spirit of  this and that's i'm just going to write  that down here on in the algorithm  can you guys hear me still  somebody said my audio is like blipping  out  i am getting a sign or two of unstable  internet so you know uh if if you need  me to repeat anything just just send me  a note good thank you  okay um  all right so well uh oh maybe i should  speak slowly if it's  not coming through too well  okay  um  so  so what we're going to do is go through  eat all the nodes of the graph one by  one  and guess whether it's a reachable node  or not a reachable node  if we guess it is reachable  i'm going to also guess the path which  shows that it's reachable  and then  i'm gonna and i'm gonna keep a count of  how many reachable nodes i found  if that count agrees with the value c  um i started with  then i know i found them all  and if t is not one of them  then i know t is not reachable  that's the idea  okay so  here's my this is going to be a count of  the number of nodes that i have found  which are reachable  um that's k  now here i'm going i'm going to  non-deterministically choose is it a  reachable node or not  i've just called it  two branches of the algorithm the p  branch or the n branch p means there's a  path  and n means there's no path  so if i guessed p at this point for this  node u so i'm going through each of the  nodes one by one each and u is the  current node  if i've guessed that  it's it is it does have a path from s  then i'm going to  uh guess that path to make sure that it  really is a reachable note  um  if i fail to find a path then this is  one of the branches of the  non-determinism  that is going to fail it's going to punt  it's going to say i don't know under  this branch um  because either  you guessed wrong and this node was not  reachable  or if it was reachable you guessed you  you failed to find a path which shows  you that it's reachable there was some  path but you didn't guess the right one  um either way you made a bad choice  you're gonna you're just gonna punt  now  if you have determined that t is uh um  that node that you've just shown is  reachable because at this stage you did  not fail so you succeeded in showing a  path to  um  then um  uh  and u equals t then you know t is  reachable that so there is a path from s  to t and you're finished  that you know you've got the answer  you're looking for and so now you can  say yes  otherwise if t is some if you is some  other node then you can just increase  your count of the number of reachable  nodes that you found  okay so you've found a reachable node  if it's t you're great you're done if  it's not you just include that as your  in your account of reachable nodes um  now if you've guessed that the node is  not reachable okay then you just proceed  you know you  you're not gonna  you're just gonna move on to the next  note because you're looking for a  collection of reachable notes  um  okay uh i'm getting some questions here  but let me let me wait till the end here  um now  after i finished going through all of  the nodes  so i'm finished with this uh um this  loop here of going through all the nodes  um  now i see did i find  c reachable nodes because k is the count  of the nodes that i've found to be  reachable if that  agrees with c then i know i found them  all if it differs from c  then something has gone wrong  because i am told there are c reachable  nodes and i did not find c reachable  nodes  so i made some bad guess along the way i  guess some node which really is  reachable  i guess it was not reachable so i didn't  find them all  i'm gonna punt  um but if i found them all  and so and and i didn't end up accepting  it it didn't say yes at this stage so t  was not one of the ones i found a  reachable then i then i  uh i'm convinced that t is not one of  those that are reachable that was not  one of those c nodes that i found which  are reachable and now i can say no  okay so let me let me take  questions here uh because i think we're  yeah that's the end of this slide this  is this is the kind of  an important  uh  piece to understand  we can spend a couple of minutes trying  to work through this  uh  so somebody's asking how does  non-deterministically pick a path fail  it if you fail  um  what i mean is pick a path from s to u  so you have to go from s to whatever  your current node u is  so you're going to pick some path  to u  you guessed u is reachable now you have  to demonstrate it's reachable by picking  a path from s to you if you don't end up  at u  um  and the pair you don't want to go  forever on any branch so you're going to  limit it to m steps your path has to be  of length m at most  so after m  steps if you have not reached you by  that point you've picked a bad path  and you're going to reject  um  okay uh  okay so what's the difference between  know and reject that's a good question  um  reject in this case is an i don't know  the algorithm  uh could not make a determination based  on the guesses that it's made  in this non-deterministic branch of the  algorithm it made bad choices which  doesn't allow it to reach a conclusion  one way or the other  okay  remember this algorithm here is  computing a function now  it's not a now it cannot deterministic  algorithm in the language recognition  sense this is a function computer and so  it has to get the answer to the path  function which is a yes or no or i don't  know on some branches or some branches  that's allowed to do that too  so no and reject are totally different  um  okay  okay this is the same thing we talked  about last time why do we need two  branches for p and n um  if we're only going to have  proposal just to have the p branch  well but some nodes are not reachable  if you're gonna look for  if you're gonna you know  if you have a an unreachable node  so it's not an r  you can't get to that node from s  you have to skip over that note  because you're trying to find a subset  of the reachable nodes  so you're trying to pick that subset  here um one note at a time  so if you're only going to allow things  you're going to require everything in  this subset there are going to be some  nodes which are not reachable and you're  not going to find a path because they're  not reachable and you're going to end up  projecting all the time  you know on that node so you're going to  be this the algorithm will will it will  not will not work  um  okay so i'm not sure i understand this  question here but somebody says if t is  reachable we output yes on that branch  but don't we also output no  on some other branch  let's that's that's a good cool let's  see what happens  if it's  if t is actually reachable  um  how can we up so if t is reachable  there's some branch that's going to  output yes we all agree with that  at least if you're following we agree  but  how could some other branch output no  if t actually is reachable  because that's a great question  um and no that's not going to happen  uh  if t is actually reachable how could a  branch output no  um  [Music]  that must mean  that it does not guess  t as one of the reachable nodes because  it's going through all of the nodes here  um you know it's going to all the nodes  and it's picking them as reachable or  not  if it pick t is one of the reachable  ones then it's going to output yes  because it will find you know it'll  either output yes or if it doesn't find  the right doesn't guess the right path  it'll end up rejecting on that uh on  that path but some path will will end up  saying yes so if t is reachable  and  uh if you guess if you know t is  reachable and you guess t  you know you guess u is reachable one at  the point when u equals t you will end  up outputting s  the only way you could not  not  output yes is if you guess that node is  unreachable  but then your count is not going to add  up right  because you you wouldn't you did not  find all the reachable nodes if t is one  of the reachable nodes and you know  there are 100 reachable nodes  and you skipped over t as one of the  ones that you say is unreachable you at  best can only find 99 reachable nodes  and and you're not going to end up  saying no you're going to end up  projecting  so it's a very good question but you  have to think through what's going to  happen here that's  this c here is kind of a check  it's almost like you know um  uh  well in  it's like a checksum if you know what it  is it's so it makes sure that everything  that you if you got to see  if you got the if  if k equals c at this point that means  you actually found all the reachable  nodes so c is kind of a check  that you found all the reachable nodes  um  right so  if k equals c at this point you have  found every reachable node  and if t was one of the ones that are  reachable you found t  uh  okay let's see  um  is the reason we do this with c  essentially so that we know and we can  stop guessing and correctly identify if  it's impossible to reach t  well it's not a matter of it it's not a  matter of stopping guessing it's it's  it's a  it's a check that we found everything  because we're going to go through  and do all the guessing for every node  no matter what  so we're not going to stop anything  early unless we find that t is reachable  then then we can stop early but if to  show that t is not reachable we have to  go through the whole process  um  how can we intuitively see that we don't  have contradictory branches that's sort  of i was trying to say that just now i  don't know i i hope that got through  you can't have contradictory branches  because if you got to this stage here  you have found all the reachable  nodes so you've at this stage if you got  to six  you have made all correct guesses  you have found all the reachable nodes  you have convinced yourself that they're  all reachable um by  guessing the pairs to them and you've  checked that you have the right number  of reachable nodes because it equals c  so you must have found them all  so you cannot have a contradictory  answer because either t was one of the  ones you found in which case you would  have already said yes  or otherwise you found them all and t  was not one of them and so you're going  to say no you can't have both both  things cannot happen okay let's move on  so the next thing we're going to do is  the next slide is exactly the same as  this slide  except instead of saying  is um  is t reachable i want to know is it  reachable within d but within distance d  okay so um  which is going to mean exactly the same  procedure can i get  uh  instead of asking can i get  um  um  uh  from s to t  with a path of any length of course it's  going to be most length m now i want to  know can i get to it from s to t by a  path of most length d  these are number of edges in the path  say  um  and uh  that's the same procedure because  instead of i'm just going to cut things  off at  a d  but if i if i if i know in advance how  many nodes are reachable within d  um  i'm going to find all the nodes that are  reachable within d and c was t one of  the ones reachable within d  it's the same exact idea so here here is  the next slide which kind of shows that  uh so here here's the definitions  path sub d means reachable  by a path of length of most d  okay so um  r sub d is all of the ones that are  reachable  by a path of that length  and c sub d is the count it's a number  that are reachable uh within d  um so if you understood the last slide  hopefully this slide will seem kind of  obvious to you i'm gonna just highlight  all the changes  so if i can now calculate  c sub d which is the number reachable by  a path at most of length d  then i can test whether or not nodes are  reachable by a path without length  first i calculate c sub d  i go i pick every node as being  reachable within d or not  now i just have to check that my path  that i'm guessing has a length of most d  instead of length at most m which is  what i had before  keep a count of the ones that i found if  that count  equals c sub d then i know i found them  all if it's not equal to c sub d  then uh i've made some bad choice along  the way and i can just punt and say i  don't know  um  and if t was not one of the ones that  i've shown to be reachable within d  then i know it's not reachable within d  and so um  uh i can say no  okay so i don't know if this merits any  additional questions um but this is  really the same  it's just a repeat of the previous slide  what's kind of amazing  is now  the last slide  is  going to be again a repeat  uh let me just let me just force out of  where we're going but feel free to ask a  question on this or on the first slide  if you didn't on the previous slide if  you didn't get that um also we can try  to help you out with that  um  the next slide what i'm going to do  is show  how to compute  all these c values  and i should mention um  the value c which is the total number  reachable is going to be the same as c  sub m  reachable with an m the number of nodes  of the graph so if i can get up to c sub  m i'm done  um  and what i'm gonna show you is that  uh  knowing c sub i i can compute c sub i  plus one or c sub d i can compute c sub  d plus one  since i'm using d as my index here  basically  so c sub zero we know is just s well r  you know it's just one because you can  read just  start with s that's the only thing  reachable with zero  and then once i  know that i can fi figure out c sub 1 c  sub 2 c sub 3 and so on and then i get  the c sub n and then i have the count of  the total number reachable and then i  can test the path function  um  okay so the trick now is being able to  count  uh  given c sub d i would like to figure out  what is c sub d plus one  now how am i going to do that what i'm  going to do  is that's my goal what i'm going to do  is  something in between  i'm going to do a theorem just like this  but instead of  given c sub d instead of computing paths  of d i'm going to compute paths of d  plus 1.  so  knowing  how many are reachable from d  i'm going to give a test for whether  things are reachable within d plus one  and the fact is that's easy  because this thing already tells me how  to compute whether i'm reachable within  d  and being able to be reachable from  within d plus one means i have an edge  from something that's reachable within d  so if i can figure out which are  reachable within d  well and i just want to say do i have an  edge  you know  uh  do i have an edge from from one of the  nodes that are reachable within d then  i'm reachable within d plus one  then if i can test whether individual  nodes are reachable within d plus one i  can count  how many nodes are reachable within d  plus one that was that very first easy  theorem that i showed  so i know there's a lot of pieces here  that you have to put together but in the  end each p each individual piece is not  that bad  um  okay i don't know how many of you have  followed me oh no this is not supposed  to be here uh  there we go  so here is the last part  which again is just a simple  modification of what what the previous  slide had  so  i'm going to show how to compute  the path d plus 1 function so testing if  there's a path of length d plus 1 from s  to some node t  but only knowing how many nodes are  reachable within d  so i'm going to find all nodes that are  reachable within d  just like i did before but see if any  one of those nodes has an edge  to t  not necessarily that one is equal to t  because that says that t is reachable  within d but i want to know does it have  an edge to t that means t is reachable  within d plus one  so if i find all the nodes that are  reachable within d  and t turns out to be reachable from one  of those with it by an edge  then t is reachable within d uh d plus  one  and if d is not reachable from any of  those nodes with an edge then t is not  reachable when d plus one  i hope you're following me i'm not sure  you are uh  so anyway that's the that's the  algorithm here and  uh  uh the corollary is that you can compute  c sub d plus one from c sub d because if  you can count the path you if you can  test for each node if it's reachable as  i mentioned before you go through all  the nodes see whether the reachable and  d plus  d plus one and then  count them up  now i have c sub d plus one and now i'm  done  because  uh  uh  i'm gonna  compute each d plus one from the from  the value d that i previously computed  i'm going to do that from for all these  should say zero here actually um  and except  uh if the path says if this if the path  function now says there's no that the  answer is no  because i'm trying to do  you know the complement of the path  language  and reject of the path uh thing for m  says yes  and that's my non-deterministic  algorithm for the path  complement  uh problem  anyway  maybe you need to look at a little bit  offline um  it's presented in a little bit different  way in the book uh i don't know if that  will be more or less clear to you but i  kind of i think this has been a little  bit more unpacked for the purposes of  the lecture  um  so let's just see do we i'm not getting  any questions which probably means  uh  uh i've lost a huge chunk of you um but  uh the good news is we're gonna move on  to a different topic  so  um  but feel free to ask a question on this  if you want  or we're going to shift gears now to  talking about the hierarchy theorems  which is going to be the second half of  the lecture  also not so easy i have to say  probably a little less technical than  this one is but  it's also this are going to be  um spending time on just mainly just one  theorem  but anyway so looking ahead to what  where we're going and then we'll have a  break  um  we uh  well what we've shown so far  these are the major complexity classes  i'm not not including let's say the the  complementary classes the co-np uh type  classes um  these are the major complex classes  we've seen so far  and as we've seen they form a hierarchy  of containments  some of those containments trivial and  some slightly less trivial  but we have not shown whether any of  these classes are different  you know we've pointed out that there  are some unsolved problems here but do  we know any of these classes differ from  each other  or could it all collapse down to l  and the answer to that is  we do know that p space and l are  actually different  that we can prove  and it's a um  it relies on the theorem  that says if you give a  turing machine more space  then you can do more things  so because p space  is much is a bigger bound than log space  we know we can do more things  in fact  because  n l is contained within  log squared space deterministically and  p space is bigger than that we actually  can separate p space and nm  so we're going to prove that today  um so the basically the idea of the  theorem says that if you give a turing  machine  a bit more time or a bit more space then  it can do more  though you have there are some  conditions on that that we have to uh  we'll get into  one of the conclusions that we'll show  is that  um  time n squared if you compare with time  it's you know the time com the things  you can do in n squared time uh versus  the things you can do in cube time there  are more things you can do in n-cube  time then you can what you can do with  n-square time i mean that's what you  would expect  um but  it's not the case that everything we  expect in uh complexity theory we can  prove this is one of the things we can  prove  so as you add more time you can do more  things so this is a proper subset here  so there's there are some things in time  n cubed that are not in time n squared  and ditto for space  okay um  so that's going to be  that brings us to our  coffee break  and so feel free to shoot me any  questions about  what we've done so far or anything else  and otherwise  we will um  launch our timer and i'll see you in  five minutes  okay so i get getting some good  questions here could we also make a  solution this is getting back to that um  um  uh  um  the logs the nl equals coin l somebody's  saying could we just  um make another selection just by  non-deterministically choosing c  vertices  and then not and then checking that  they're all reachable  that's effectively what we're doing  but  be careful that we because we cannot  store c vertices  so that's why we're doing them one at a  time  we can't guess all c vertices up front  because where you're going to store all  that we only have log space  okay another so maybe somebody's asking  how much working space do we need for  storing the intermediate steps i'm not  sure what intermediate steps you mean  but if it's all the ci values  you know see  going from c0 to c1 to c2 to c3 we don't  store those  we all you need you need c sub d  to calculate c sub d plus one and then  you forget c sub d you couldn't store  all the c values but you don't need them  all you only need the most recent one to  go to the next one okay somebody's  asking can i go over why the complement  of path in nl  implies nl equal coin l  because the complement of path  is  uh essentially it's co n l complete i  mean l it is um  so  everything in nl is reducible to path  the  everything in cohen l  is reducible to the complement of path  by the same reduction  um and so if you can do the complement  of path in any uh class you can do all  of the complements of nl languages in  any class and so you can do the  complement of path in nl you can do all  the  connell problems in out in  nl and so then nl equals comma now  you have to you have to just think  through the logic of it this is not that  that part is not hard um  why it's enough to solve the path  complement problem in nl that does  everything else because it's a good so  the same completeness phenomenon that  we've been seeing  okay somebody's asking about the the two  sat problem that we talked about last  time um and is it um  and you know i i pointed out that that's  in  nl the you know the the two side problem  well the complement of the two set  problem  uh  you know the unsatisfiable two set  formulas that that's an nl language  because you can basically look for a  contradiction uh non-deterministically  in log space  um  that's  i think i probably won't be able to  explain that in a minute but maybe we'll  have our recitation instructors cover  that in recitation  um it's a nice proof not very hard but  it's it's an it's a nice proof  it's it's it's it's it's you know it's  not  it's something you have to do you have  to think about you you have to argue but  it's still it's not super hard um  understanding the two sad problem and  you know the complement of two set we  showed is in nl  and because nl equals coin l also the  two sat problem itself without  complementation is in nl  and in fact is nl complete  okay  um  i think we're going to have to move on  uh i'll stick around after lecture in  cases any questions that i can answer  quickly at that point  sorry if i couldn't get to your question  just now  all right  continuing on here  okay  shifting gears the space hierarchy  theorem  so  um  so as i mentioned i think um  uh maybe it's good to just go back  to this slide here um  we're going to do the time and space  hierarchy theorems which show that if  you can do a little bit more if you give  a little bit more time or a little bit  more space you can do more things  we're going to do the space  case  first because that actually tends to be  slightly  for certain technical reasons slightly  easier  so  uh  space hierarchy theorem so here is  the statement of the theorem and  it says for any bound  think of s is going to be some  you know  space bound  and again  f has to satisfy some technical  condition in yellow remember that it's  yellow because that's going to be  relevant later  um  uh so there's going to be some technical  condition  um  for no matter what function you you have  whatever space bound you have as long as  it satisfies this condition which is a  mild condition but you need it uh  whatever space bound you have um you can  find a language a  which requires exactly that much space  so if f is like n cubed we're going to  find a language a that requires n cubed  space if it's n to the hundredth we can  find the language that requires into the  100th space  and cannot be done within 1999 space  whatever it is  you can find a language that requires  exactly that much space and if you like  it a little bit more formally so that  means that it can be decided in that  much space  but it cannot be decided in less space  okay  framing it in a slightly different way  in terms of our space classes  um i'm going to define a notion which is  you know kind of it's not  it's  not said this way in the book but maybe  it's a helpful way to write it down  it's space little o of f of n  so those are all the things that you can  do by a function that's little o of f of  n  in space  so space little o of f n is properly  contained within space f of n  in other words there's something here  which is not in there  okay picture pictorially  i'm going to exhibit some language some  explicit language a  which i can do in this much space but  not in any less  now you know you can sort of think of  this as a little bit like the situation  for context-free languages and regular  languages where we exhibited a  particular language that differentiated  that was it context-free but not regular  and we're going to kind of do the same  thing now  but the one key difference is that in  the case of separating the context free  and the regular we could give a nice  language like 0 to the k1 to the k  here the language is not going to be so  nice to describe  it's going to be the language that some  turing machine we're going to give  decides  but you're not going to be able to get a  nice simple understanding of a it's  going to be whatever that turing machine  does and so in that sense it's not a  very natural language um that's easy to  sort of get your mind around um  so the i outline and really you don't  have to  worry about this but maybe it helps it's  really going to be a kind of a  diagonalization proof  um  the way this machine d  um is going to operate so d is going to  give you my language a  um  so d is going to be designed and i'm  going to show you d on the next slide  uh d is going to run within my target  space bound f of n  and here's the key here's the kicker  d is going to be designed to make sure  that its language cannot be done in less  space  and the way it does that  is it makes sure that its language is  different from any language that  that is um  decidable by a turing machine in in less  space and it's going to be different in  at least one place  so any  d is going to guarantee that its  language cannot be done in little o of f  of n space because it's going to be  different from every language that's  doable in little o of f of n space  somewhere  okay that's the point and then the  language a is going to be  the language of this  uh turing machine d  okay so it looks like a tall order the d  has to make sure that each  you know that that for every machine  its language differs  uh from that machine's language if that  machine is running in little of f event  space  but it's basically going to be a  diagonalization  so  for all of the different possible inputs  to d  that input is going to actually code up  a machine  uh on which  we're going to make sure that we're  different from that machine if it's a  small space machine  okay  let's see  so i can take a couple of questions here  does f have to be computable so that's  going to be one of the conditions um  that we're going to have to guarantee  where f satisfies the technical  condition yeah it's going to end up  being half it's going to be computable  but that's not enough  um  good question though  uh  okay so let's let's  let's move on from there okay now  so here is now what's what my job is to  give you this turing machine d  so d  these language is going to be my my  language a which i can which requires f  of n space cannot be done unless  okay oops i need to  i need the full slide here so i have to  take myself out  um  [Music]  all right uh now  this is my goal i want to exhibit this  language a which i can do in this much  space but not in any less  and  so i'm going to give this machine d as i  mentioned where a is d's language  d runs in order f event space and that  sort of that achieves this part  and d it makes sure that its language  cannot be done  in any less space  so that achieves this part  so it's different from the language of  any machine that runs in little o event  f of n space  okay um  [Music]  so uh  this is how d is gonna work  okay i'm gonna try to give you a little  picture uh to help help see the to  accompany the description  uh so d  gets its input w which is of length n  the very first thing d does  is it marks off  f of n's space  because it's only allowed to use  we're only going to allow d to use f of  n space because otherwise we're in  danger of  d  not  of a not being in in space f of n so d  is going to guarantee that by making  sure  it's going to mark off f of n space  and if it ever tries to use more than  that it just rejects  and but  by virtue of that we're sure that these  language is in space f of n because d is  an f of n space turing machine and it's  going to be the side  okay so  this part  so far is not too hard  okay now we're going to start getting  into the meat here  so  if w  um  now what we want to think of w as a  description of a machine that we're  going to feed that's that's going to run  on w so this is going to a little bit  you know  back to an earlier when we talked about  diagonalization so don't get  thrown off by this um  we if  we're going to think of w not only as  the input to d but it's also going to be  the description of a machine  and if it turns out that w is doesn't  describe anything it's just a jump w  then we're not not interested we're  gonna we're just gonna reject on that w  we're only interested in the w's that do  describe some machine m  okay  so if m  if w  uh describes some machine m  then we're gonna run m on w  and we're going to do the do the  opposite of what uh what m does  that's the whole idea  we're just going to uh make sure that  what we're doing is not the same as what  emma's doing  so at a high level  the the basic idea for this  is is not hard um  so we're going to simulate m on w  if m x rejects then we'll accept and if  m accepts then we'll reject we're just  going to do the opposite  um  and i think that is  so we have to be careful when we do the  simulation this is a little bit of a  detail but you know  this is a proof where you need to pay  attention to some details  um  the cost of simulating  m  on d  is only a constant factor  um because if m uses a certain amount of  space  when d is simulating m you know m may  have a larger tape alphabet than d does  but these can then encode um m's tape by  using several cells for each of m's  cells but it's only going to be a  constant factor and that's important  here because  um we have to make sure that  you know if this was a big blow up um d  would not be able to run m  um  i i think i'm sort of arguing the  details without making sure we  understand the fundamental concept um so  let me back up  the point is that d is doing something  the opposite of  m now  uh d can't be different from every m  because d itself is a turing machine of  course but but the thing is is that  d is only running within f of n tape  cells so it has to be able to do that  simulation of m within that amount of  tape if m is using a lot of tape then d  is going to use a lot of tape and it's  just going to reject  so this is only going to really come  into play  getting being able to simulate m if m is  using a small amount of using a small  amount of space so that d can do the  simulation  okay so let's just see maybe  uh  so they're going to be some issues here  but before i get to that let's just see  what uh what your questions are  how can a turing machine know if w is  encoding some other turing machine  no that's simple  you know what what is a coding of a  turing machine it's just you know the  standard we have a standard coding um  it's just you know coding the rules of  the machine so it has to have states  transition function blah blah blah so it  just has to be some you know whatever  our encoding for the turing machine is  we can always test whether  a string is a legitimate encoding of a  turing machine so that that shouldn't be  bad um  somebody says why do we reject if we use  more than f n cells isn't it okay to use  order f of n  yes it could be but  we have to cut it off somewhere  you know  it might be it's okay we could use two f  of n we could use 10 f of n but we have  to have some constant for d  and let's just kind of simply constant  one so d has to run within f of n cells  um and that's going to guaran that's  going to be good enough for us  okay do we have to make sure that m runs  a little of f of n so we can't really  tell  whether m is running in little o of f of  n  or we can tell us whether we can finish  the simulation  so that's actually going to be  maybe you can just hold off on that  question because there is a point that  we have to follow up on in that which is  um  uh just because  you know we may or may not be able to  finish simulating m  on this w  doesn't necessarily tell us what the  asymptotic behavior of m is but we'll  have to look at that in a bit okay so  somebody's saying what happens if m  loops on w that's going to be one of our  issues we have to deal with that's a  good question there step two alone can  use them more than f of n cells  yeah step two alone can use more than f  of n cells if it does we're just going  to end up projecting  okay so we're getting good questions  here some of them we're going to which  i'm going to address anyway so why don't  we just move on um  okay  so here is sort of a question i think  this is one of the questions that that  related to one of the ones that got  asked what happens if it runs in little  of f of n space so we we remember what  we're trying to do is be different from  every small space you know little o of f  of n space machine  so what if m runs a little o of f of n  space but has a big constant so what i  mean by that concretely is suppose d is  an n cubed space so suppose we're trying  to  get a  in in cubed space  but show what's not in n squared space  d is going to run an n cubed  and what and we have to make sure that  any machine that's running in n squared  space cannot do the same language  um so we're going to be different from  that but the problem is that uh in  uh  the machine m  might be running in  n squared space but with a huge constant  so it might be running in a million n  squared so that's still a machine that's  running in  a little o  of n cubed and we have to be different  from it but for the particular w we're  working on we might not have enough  space to run m because of the huge  constant it that con  the asymptotic behavior is only going to  be relevant well for large w for smaller  b we may not see that we may not have  enough space to run m so  um  what are we going to do to fix that  we're going to run that m on infinitely  many different w so it's going to be  infinitely many different w's  that are all going to encode the same m  and the way i'm going to do that  is by  uh  thinking of w as representing m  but having  an unbounded number of trailing zeros  after that  so i'm going to strip off the very first  thing i'm going to do with w is i'm  going to strip off the trailing zeros up  into the final one i'm going to remove  those  and then take the rest  and as the description of the machine  so now i'm going to have potentially  w's that have an enormous number of  zeros at the end  big enough so that i can see the  asymptotic behavior of m  and that if m is really running in  little o of f of n space i'll have  enough space to run m to completion on w  and so then i'll be able to be different  from it  okay  um  so i'm kind of showing that over here so  here's a very large w  i'm going to strip off the trailing  zeros  the rest of it is just going to be m and  i'm going to run this m on the whole w  the entire w without the zero stripped  off so now m is going to be running on a  very large input  um  big enough so that d uh d which has  asymptotically more space than m does  will have enough space to run empty  completion  um  now another question that got asked what  happens if m loops  that's going to be a problem because d  always has to hold  and if it just blindly simulates m  then d might be looping on m  none of m is going to use a lot of space  by the way  because then d is going to catch it in  step one but if m uses  his loops on a small amount of space  then uh d  might end up looping as  presently constructed so what i'm going  to do  is i'm going to put a counter which  makes it stop if it runs for  2 to the f of n space so basically  because that's how long d could possibly  run without looping anyway  m could be running without looping  anyway and so we're going to run it for  this amount of  this number of steps and  uh i'm going to reject if it hasn't yet  halted as well as uh that because it has  to be looping at that point anyway um  and so it's not interesting for us  it doesn't matter what we're going to do  if it hasn't halted because  m is not a decider  and the last thing is how to compute f  um  so i'll try to address some questions  here in in our remaining time  uh how to compute f  so to mark off f of n cells we also have  to compute f i didn't think anybody any  of you guys asked that question except  maybe sort of the very beginning about f  being a computable function  certainly f is going to have to be  computable but not only does it have to  be computable it has to be computable  within the space bound  and that's just going to be a condition  we're going to impose on f it's so  called space constructable namely that  you can compute it within its own space  bound  and all nice functions that we care  about are going to be space  constructable  so  it's not doesn't turn out to be  an obstacle to applying  uh the hierarchy theorem but it is a  condition that we need it actually is  not true without that condition  um  okay let's let's just oh this i have a  check in here maybe we can take a couple  of questions first  some of you are anticipating my check-in  actually which is good um  so let me hold off on those  sorry a bit confused about what is m  can we say d as input m and simulate m  on yeah so  uh somebody's saying  can we say that d has input m  and simulates m on itself yes that's  exactly what's happening the reason why  we're doing that is because we have to  cover all possible m's  so as we get all possible inputs w  they're going to range over all possible  ends and so every possible m is going to  get addressed to see if we can run it um  within the space bound and be different  from it  d's job is to be different from each of  those ends  but it's not you know again there were  some details here that got raised in  these issues um  but in a sense this is just kind of  more technical  i would focus on understanding what i  originally wrote down because that's the  main idea the rest of it is just kind of  implementation details  okay so why don't i  can i give an example of a non-space  constructable function  yes  log log log n space you cannot compute  log log log n space within log log log n  space  and in fact it's known  that  there's nothing new  between constant space which is just  regular and log log log n space anything  you can do in log log log n space is a  regular language  so the the hierarchy theorem doesn't  doesn't won't apply there because well  it applies but that's not a it doesn't  well you know it's not it's not space  constructible  to find  higher level um you know large  um  non-space constructable functions you  can do it but they're  you know you you  they're not easy to describe  okay let's do our check in here  what happens when we run dion itself i  got a couple of people asking me about  that so that's a this is just a good  lead into our check-in and this a little  you really have to understand what's  going on  to see what does d do  when if you feed in itself maybe with  some trailing zeros because remember the  algorithm strips off trailing zero so  what is what does it do in that case um  so here are my  options there you can get to pick which  ones which one  you uh you think is the answer so i'll  give you another 30 seconds on this  because this requires a little bit of  thinking  if you want to invest in it all right  i'm wrapping this up guys  five seconds to go  okay i'm gonna end it so get your  participation points in  a bunch of you have not uh said anything  come on  um i can see the count here and it's  there's a  three or four of you are not not  answered well you're going to lose out  closing  all right so the right answer is in fact  c it does reject let's just understand  what happened is definitely we don't get  a contradiction i mean this is an  algorithm i just described it's going to  do something  um  uh  i'm assuming the people who picked e are  having fun as i did when i came up with  the check-in but um  uh not a question answer a doesn't is  not going to be good either because d  has to be a decider so it can't loop on  anything so the only sort of reasonable  answers are accept or reject  when you run d on itself  uh  what's it going to try to do  it's going to the very first thing it's  going to you know mark off f of n tape  cells  and then it's going to get its input  which is itself  tries to simulate itself on the same  input  that uh simulated d is also going to try  to mark off f of n tape cells  but um due to some simulation you know  there's going to be some cost to doing  the stimulation when the simulated d is  going to try to mark off f of n tape  cells it's going to blow  the original d  space bound and exceed the bound and so  d is going to reject up right in step  one  when it tries to get an input of its of  itself  so that's um  it's very clear what's gonna happen in  this it's just gonna reject  because of the for this reject this  reject in particular  and notice this would you know yeah  you know  okay let me not try to confuse it um  okay so that's that's that's all i want  to say about this um let's now move in  our remaining seven minutes to the time  hierarchy theorem which is very has the  same proof um but some of the technical  details are slightly different  okay um  so now if i give you a time bound again  we you have gonna have a face with the  same notion that you have to be able to  compute f within f's amount of time  so it has to be a time constructable i'm  not gonna define that um  so there's a language a which requires  that much time  um so it has to be decidable within that  much time but there's a slight  difference here and there's and this is  an artifact of the proof of the theorem  not because it's an absolute truth as  far as we know  uh it's not that it's not decidable  little o of f n you actually you can  only prove something slightly weaker  when you have one tape turing machines  that it's little decidable little o over  there's there's a slight um  uh  gap in what you can prove so it's not  only that you can't prove you it  requires little o but little o of  little of f of n over log f of n  is uh what you can prove that the you  get from this uh time hierarchy theorem  but let's not get caught up on that for  now um  uh  okay  so the proof outline is the same outline  as we had before um  uh  we're going to give a d that runs in  order f of n time  so it's  ensures that the language is in that uh  time complexity class time f of n  and it makes sure it's different from  every machine that runs faster  by some significant love by a log factor  faster  okay and so um why don't i  uh  show how that goes  um  the proof is in some ways  almost exactly the same  um  i'm going to give a d  which runs this much time  and it shows it's different from every  m that runs in  a lot less time  here is the algorithm for d  now it computes f n but it does  something a little different with f of n  remember in the space  hierarchy theorem  we marked off f of n space  now this f event is going to be used for  a different purpose it's going to be a  clock  and you have to shut m down  if it runs for more than f of n steps  not if it uses more than f of n space  because we're only interested in m's  that use  significantly less  than f of n time  so we're going to run an m you know for  f for some number of steps  whatever m says we're going to do the  opposite and only if we can actually  finish that simulation we'll be be able  to be sure that we're different from  what m is doing  um so this is the whole algorithm here  we don't have to do any further  modifications  and where's that login factor coming  from  it's actually coming from a funny place  um  and and you know you have to get into  the little bit of the guts of this  when you're simulating m on w  remember that m itself was described by  w  so you're going to have to write down a  copy of m  which is you know just  as described by w  and then so you're gonna and then you're  gonna have the tape that am is working  on  which is starting out with w on it  um  and uh  you have to be now you have to be a  little careful how you manage that  because if your description of m is just  sitting at the beginning of the tape  as you're simulating m  every time you do one step and modifying  the tape  you don't want to have to go back  to the beginning of the tape to look up  the next step of m  so you actually have to carry m along  with you  as you're doing the simulation  and you can do that by expanding the  tape alphabet of the tape  so that you can effectively have two  symbols  on  one cell one is going to be  for describing m and the other one is  going to be for just the for m's  for the simulation tape  and you'll be carrying m along with you  uh wherever your head is  so you don't have to go very far to look  up m  and  so that's all possible because that's  going to add only a constant factor  because m is fixed in size doesn't  depend on  you know  you know  for large inputs to m m is fixed  but the tricky thing here is  the counter to make sure we're not using  too much time  um  [Music]  the counter has size log f of n because  that's how big it has to count up to so  you should you can shut it down if it's  going to exceed the f of n steps  um  and  because you know you have to  run for a certain amount of time  and  carrying the keeping the counter nearby  has the counter now could be pretty big  and so that's going to cost you a log  factor of um  of simulation cost  to  move that counter around all the time  and so  that's why you have to  run for only a log factor less  so that you can actually finish within f  of n time as you're as you're required  to do  okay i realize that that's a mouthful  there and you may not have all  understood that  it doesn't matter it's not that critical  i think what i'm really more concerned  is you understand the the main idea of  the um the hierarchy theorem some of  these implementation details  you know you  if you don't get them  i wouldn't worry about it i i feel i  have to include them for completeness  sake and to be honest with you about the  proof but if you didn't follow  everything that's okay i do want to  understand the main idea though of the  algorithm  making sure that what it's doing is  different from what every machine is  doing if that machine runs in little o  of f of n space or or a small amount of  time you know little o of f n over log f  of n time  okay so and i think we're gonna uh  we're gonna end here so come pretty much  out of time uh i'm gonna stick around  for a little bit  um  in case there's any questions here oh  wait there's one let's check in though  um  let's let's look at this this is kind of  an interesting sort of follow-on to the  hierarchy theorem um if you look at the  two questions does l equal p and does p  equal p space  these are both unsolved problems  does the what if anything does the  hierarchy theorem tell us about those  questions  and it's kind of interesting that there  actually does well i'll leave it to you  to  uh  tell me if you can see what what it  might actually be telling you closing  get your answer in  okay one two three  i feel like i'm running an auction house  here  i should have a gavel  okay  yes in fact we know that you know these  are separated so it's not even though we  don't know if l equals p or p equals p  space they can't both be equal  because then um l would equal p space  and we know that's false  um  so at least one of these has the answer  no  okay so with that let's wrap up today's  lecture  you know we prove these hierarchy  theorems and why don't we just uh  um  i'm going to  uh  shut us down here but but before well  we're over so you can feel free to go um  but i'll stick around in case anybody's  any questions for a few minutes anyway  um and then uh  we'll call it a day  since we just showed space and is a  proper subset of space n to the k for  any k why can't we also say  space n is a proper subset of p space  yes space n is a proper subset of p  space  yeah so somebody just asked we just  showed that space n is a proper subset  of space n to the  k does that also say that space n is a  proper subset of p space definitely  any  um  space n to the k is a proper subset of  space n to the k plus one  which is a subset of p space so any  fixed polynomial is going to be a subset  of p space because p space includes all  the polynomials  which of the two unsolved problems  uh whoops  do i think is more likely likely to be  true  well  i think most  i mean i would bet that both of these  are not equal so both of these have  answered no  um  uh it would be weird you know  i mean you think l equals p  that anything you can do in polynomial  time you end with log space log space is  incredibly weak  and and p space is incredibly strong  um  i would be shocked if either of these  were equal  so we just  the problem is that we don't have a  method  for proving  um  problems  are are actually  have high complexity of of any sort we  don't know how to show things outside of  l don't know how to show things  outside of p  except by using the hierarchy theorem  diagonalization is the only method that  we have for showing things or outside of  classes  and there's reason to believe that as  we'll this will get to i think next  lecture in fact  there's kind of reasons to believe that  the hierarchy theorem type argument  which is diagonalization is not going to  answer those kinds of questions um so we  need a different method and  diagonalization is all we got  good question though  if i didn't get to answer your question  you have a question for me ask it again  because it's got buried  so it means we are very far from  disproving p versus np is that right  it could happen tomorrow  you know how do you how do you how can  you tell it doesn't you know it it seems  clear  that the present state of mathematics as  of right now  is  you know we don't have a clue  how to answer those kinds of questions  and it's not obvious that we've even  made any progress  um but  you know  that's the nature of the game that's the  nature of the beast  you know somebody gets a good idea and  all of a sudden lots of things uh can  change  and that can happen and then in any  point maybe one of you guys  when were these results first  this is the the stuff okay the hierarchy  theorem is old  that goes back to the very when time  when time classes were first defined i  think is one of the first results to  show the hierarchy  and  that's  late 60s  the  nl equal coen l i think i mentioned was  like mid 1980s  much later  i mean from your points of view it was a  bit it was back in the  cave  cave age uh either way but uh yeah but  but the the hierarchy team that that's  actually pre predates uh  my coming into the field but but the but  the nl equal conel that was something  that i was  i personally experienced how surprised  people were  okay i think i'm gonna  send you off all off on your way good  good having you here and i have a good  weekend everybody  and um  i will see you on tuesday  bye  you

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okay  welcome everybody so we are um  here today lecture number 21 uh coming  into the home stretch of the course  uh i'd say probably  um this last quarter of the class of the  course is a bit more technical uh  perhaps so a little bit more abstract  some of the theorems are going to be  more difficult um so  i'll i'll try to work through them  slowly and answer your questions but uh  um  you know it's i think you can expect  them to find the material a bit more  challenging as we started uh  um  [Music]  uh  as we started um  um  you know with uh  this theorem last time uh  uh  uh  non-deterministic log space being closed  under complement so nl equals colonel  we kind of only got about part  maybe a third of the way through that so  i'm going to start over with that and  spend  kind of the first half of the lecture  today talking about that and then we're  going to talk about the  hierarchy theorems which are very  important um  uh aspect of the complexity landscape  basically they tell you  that if you allow  you know your favorite model let's say  turing machines to have more resources  then they can do more things  um but we'll get to that in due course  okay so let us  uh  go back  to oops reminder to myself  um  go back to  the immerman celeb cheney  uh which is that nl is  equal to co nl  um  so  as i mentioned these are going to be the  same slides as last time uh and i'll  just try to walk through them slowly  um i hope hope i hope you i hope you uh  get it but if you don't you know uh  ask me at the ta's questions  so we're gonna first i mean  the the the the i  the thing we're gonna show is that the  complement of path  is solvable in non-deterministic log  space we already know that path  is solvable in nl that's easy to do you  basically just  start at the start node  and you guess the sequence of nodes  storing only the current node in your  log space working memory  on the on your in your logs based work  tape  you guess the sequence of nodes  on the different branches of the  non-determinism and if you ever get to  the target node t then you can accept um  but how can a non-deterministic log  space machine know  or accept the complement of path so it  would have to accept when there's no  path  um and that uh is  a lot harder  but it's a big surprise to the  complexity community that it is it is  true  um  so as we uh discussed last time we're  going to talk about  computing functions with a  non-deterministic machine and that turns  out to be a convenient way of looking at  this  so we're going to have non-deterministic  machines  that have different branches of their  non-determinism you know on some input  and they're supposed to compute some  function value  you know remaining on the tape  but because of the non-determinism you  can't imagine that different branches  might have different function outputs  well that's not allowed  all branches must either report the  value of the function  um that we're trying to compute  or they can punt  basically they can reject  and say well you know uh  i i you know it's a basically i i don't  know  so all branches can must either report  the correct answer or they can say i  don't know and some branch must at least  one branch must report an answer  must report the answer  um for and that's what it means to be  computing a function  with a non-deterministic machine  and we're going to show that certain  functions can be computed with  non-deterministic log space machines  um  in particular this path function  which sort of  incorporates both the positive and  negative  of the pair both when there is a path  and when there is an is not a path  into into the function because the  function has to answer yes when there is  a path from s to t and no when there is  no path from s to t  okay so  if you could do this you're done  because  um you can make a non-deterministic you  could make an nl machine  so if you could compute the path  function you could make an nl machine  which would accept whenever the function  says no  um  and the other cases you know  and if  the um machine that's computing the  function  uh rejects you can then you'll reject as  well  but you accept if the function says no  and so therefore you're going to be  making a an nl machine which does the  complement of the path problem so it's  uh  you know  if you can compute the path function  that would be great  so that's what we would like to be able  to do  so as i mentioned we're going to have  two other values that that are going to  be relevant to computing the path  function which is what we're ultimately  going to do  and that's going to be the number of  nodes that you can reach from the start  uh  from the the start node  in your graph and the and then  for r is is the collection of nodes and  c is the number of reachable nodes  um  so shown on this picture and here i'm if  it's  helpful to you to see it in a more form  is  you  can think of r as a function of the  graph and the start node of course but  sometimes we'll just call it r you know  when it's clear which graph and start  node we're talking about  r is the set of reachable nodes so it's  the collection u  such that  the answer is  yes and  c  is the size of r  okay  so  the way we're going to start  is kind of an easy  theorem  though this is still going to be  relevant kind of at the end um but for  now it's really more a practice with the  concept that we have come up you know  this function concept that we've just  introduced so i want to say that  the path function  with an nl machine  then i can compute the count  uh  with an nl machine  okay so understand what computing the  path function means  that you have your nl machine  and  um  every branch has to either say i don't  know which is reject or it has to have  the answer which in it's going to be yes  if there is a path and know if there is  no path  and if i can do  be able to count the number of nodes  that do have a path  the number of notes for which the answer  is yes  and this  i think if you're comfortable with the  definition  this is  [Music]  more or less obvious  because  what you would do  is you would go through the nodes of g  one by one  and test uh  using your path function  uh what the answer is  yes or no  and every time it's a yes you add one to  the count until you've gone through all  of the nodes  and then um  you have your answer which is nodes  that's c  now if the machine  that's  trying to compute the path function on  the non-determinism rejects that that's  okay you'll  computing c that branch will reject also  but when you're um some branch  has to get the right answer on the on uh  you know for it has to get the right  answer  um and so then you get the  you know  you know what's happening with that  node and so you then um  can either increment the count or you  move on to the next node i think i'm  uh  trying to say  i'm not sure if i'm making any clear by  kind of repeating myself but okay here  so you're going to start out with the  you're given the  the graph and the start node  we're trying to compute this value c  which is the number rechargeable from  the start node  you start out with you you have a  counter which you're going to set  initially to zero and you go through  every node of the graph  and if uh  the path function computation says yes  you can reach you then you add one to  the count it says no you cannot reach  you then you just continue  um  and uh  maybe i should add another line here if  if the  you know if the if the the thing that's  computing  uh rejects then you also reject um  and then at the end you output  um  so what we're going to prove is the  other direction if i give you the count  then i can answer  the question for each node whether it's  reachable or not and this is the thing  because what it's saying that is i can  give you the count i'm done  if we can get that count that's going to  be enough  um  okay so  maybe even before the check-in maybe we  should just answer any questions because  you know if if you're  stuck here then you then you're doomed  um so i think it's you it makes sense to  try to understand what's going on at  this at this  um because i think the real the real  guts of this proof is coming on the next  slide um they're  kind of the main idea so i'm happy to  take  if there's any questions about this  i'll just wait for a second to see if  you're typing away  there well why don't we go to the  check-in maybe that'll help um  not a very difficult check-in  um  it'll come up  okay just a little practice with the  concept  so  i'm getting going to give you some graph  it has nine nodes  um and i want to know the value of the  count  um  so we'll we'll assume that s  the start node here is reachable from  itself  and now what's the value of c  okay  are we  good gonna shut this down give you  another two seconds please get your  answer in  okay  uh ready set end  yeah the right answer is in fact  uh e which is six there are six  reachable nodes in this graph and that's  what the value c is supposed to tell you  is how many nodes can i get to from s  okay and what i'm saying is that if i  can calculate that  in this sort of non-deterministic  function sense so if it's so some  branches can get that answer  um  then i can use that to  test for each node  whether it's reachable or not which is  kind of a little bit of a miracle  right that's kind of surprising just  knowing how many nodes are reachable  will allow me to test whether each  individual node is reachable  um because that's there's no obvious  reason why that would be  so there's uh going to be a procedure  for doing that which is on the next  slide and here it is  um  okay so this is the key idea that we're  going to repeat  um uh  later but  so  it's good to understand um  this is this is the slide you really  need to understand  uh  okay so i'm given the graph let's assume  the graph has m nodes  um  now as i said  okay so let's just say what what are we  doing here  given that count we can compute path  so we'll get the answer for every node  in the graph  if i just know  how many you know so i'll know i can get  the answer for which  whether a node is reachable or not  if i just have if i just know how many  reachable nodes there are  um so what i'm going to do is get get  that count of how many are reachable  now i'm going to go through  um  i'm going to because let's see what's  the idea here uh before we even jump  into the algorithm  the idea is  let's say i know how many nodes are  reachable like 100 nodes are reachable  now what i'm going to do what the  algorithm is going to do is find all  hundred reachable nodes  it's got one by one but it doesn't  matter sort of conceptually it's going  to find all reachable all reachable  nodes  and non-deterministically guessing them  so it's not sure in advance which ones  they are but it's going to guess  basically a hundred no it's going to  guess some of the nodes as being  reachable  confirm that the ones that guest are  reachable are reachable and then check  to see that that number equals 100.  on some branch of the non-determinism  you will guess right and you'll end up  with exactly the right set of 100  reachable nodes and then you'll see  is t  one of those reachable ones um in which  case you say yes or is t not one what is  not one of those hundred nodes  and then um you know the answer is no  because if you've guessed a hundred  nodes and you know they're all reachable  and you know there are exactly 100  reachable nodes then every other node is  not reachable  so that's kind of that's the spirit of  this and that's i'm just going to write  that down here on in the algorithm  can you guys hear me still  somebody said my audio is like blipping  out  i am getting a sign or two of unstable  internet so you know uh if if you need  me to repeat anything just just send me  a note good thank you  okay um  all right so well uh oh maybe i should  speak slowly if it's  not coming through too well  okay  um  so  so what we're going to do is go through  eat all the nodes of the graph one by  one  and guess whether it's a reachable node  or not a reachable node  if we guess it is reachable  i'm going to also guess the path which  shows that it's reachable  and then  i'm gonna and i'm gonna keep a count of  how many reachable nodes i found  if that count agrees with the value c  um i started with  then i know i found them all  and if t is not one of them  then i know t is not reachable  that's the idea  okay so  here's my this is going to be a count of  the number of nodes that i have found  which are reachable  um that's k  now here i'm going i'm going to  non-deterministically choose is it a  reachable node or not  i've just called it  two branches of the algorithm the p  branch or the n branch p means there's a  path  and n means there's no path  so if i guessed p at this point for this  node u so i'm going through each of the  nodes one by one each and u is the  current node  if i've guessed that  it's it is it does have a path from s  then i'm going to  uh guess that path to make sure that it  really is a reachable note  um  if i fail to find a path then this is  one of the branches of the  non-determinism  that is going to fail it's going to punt  it's going to say i don't know under  this branch um  because either  you guessed wrong and this node was not  reachable  or if it was reachable you guessed you  you failed to find a path which shows  you that it's reachable there was some  path but you didn't guess the right one  um either way you made a bad choice  you're gonna you're just gonna punt  now  if you have determined that t is uh um  that node that you've just shown is  reachable because at this stage you did  not fail so you succeeded in showing a  path to  um  then um  uh  and u equals t then you know t is  reachable that so there is a path from s  to t and you're finished  that you know you've got the answer  you're looking for and so now you can  say yes  otherwise if t is some if you is some  other node then you can just increase  your count of the number of reachable  nodes that you found  okay so you've found a reachable node  if it's t you're great you're done if  it's not you just include that as your  in your account of reachable nodes um  now if you've guessed that the node is  not reachable okay then you just proceed  you know you  you're not gonna  you're just gonna move on to the next  note because you're looking for a  collection of reachable notes  um  okay uh i'm getting some questions here  but let me let me wait till the end here  um now  after i finished going through all of  the nodes  so i'm finished with this uh um this  loop here of going through all the nodes  um  now i see did i find  c reachable nodes because k is the count  of the nodes that i've found to be  reachable if that  agrees with c then i know i found them  all if it differs from c  then something has gone wrong  because i am told there are c reachable  nodes and i did not find c reachable  nodes  so i made some bad guess along the way i  guess some node which really is  reachable  i guess it was not reachable so i didn't  find them all  i'm gonna punt  um but if i found them all  and so and and i didn't end up accepting  it it didn't say yes at this stage so t  was not one of the ones i found a  reachable then i then i  uh i'm convinced that t is not one of  those that are reachable that was not  one of those c nodes that i found which  are reachable and now i can say no  okay so let me let me take  questions here uh because i think we're  yeah that's the end of this slide this  is this is the kind of  an important  uh  piece to understand  we can spend a couple of minutes trying  to work through this  uh  so somebody's asking how does  non-deterministically pick a path fail  it if you fail  um  what i mean is pick a path from s to u  so you have to go from s to whatever  your current node u is  so you're going to pick some path  to u  you guessed u is reachable now you have  to demonstrate it's reachable by picking  a path from s to you if you don't end up  at u  um  and the pair you don't want to go  forever on any branch so you're going to  limit it to m steps your path has to be  of length m at most  so after m  steps if you have not reached you by  that point you've picked a bad path  and you're going to reject  um  okay uh  okay so what's the difference between  know and reject that's a good question  um  reject in this case is an i don't know  the algorithm  uh could not make a determination based  on the guesses that it's made  in this non-deterministic branch of the  algorithm it made bad choices which  doesn't allow it to reach a conclusion  one way or the other  okay  remember this algorithm here is  computing a function now  it's not a now it cannot deterministic  algorithm in the language recognition  sense this is a function computer and so  it has to get the answer to the path  function which is a yes or no or i don't  know on some branches or some branches  that's allowed to do that too  so no and reject are totally different  um  okay  okay this is the same thing we talked  about last time why do we need two  branches for p and n um  if we're only going to have  proposal just to have the p branch  well but some nodes are not reachable  if you're gonna look for  if you're gonna you know  if you have a an unreachable node  so it's not an r  you can't get to that node from s  you have to skip over that note  because you're trying to find a subset  of the reachable nodes  so you're trying to pick that subset  here um one note at a time  so if you're only going to allow things  you're going to require everything in  this subset there are going to be some  nodes which are not reachable and you're  not going to find a path because they're  not reachable and you're going to end up  projecting all the time  you know on that node so you're going to  be this the algorithm will will it will  not will not work  um  okay so i'm not sure i understand this  question here but somebody says if t is  reachable we output yes on that branch  but don't we also output no  on some other branch  let's that's that's a good cool let's  see what happens  if it's  if t is actually reachable  um  how can we up so if t is reachable  there's some branch that's going to  output yes we all agree with that  at least if you're following we agree  but  how could some other branch output no  if t actually is reachable  because that's a great question  um and no that's not going to happen  uh  if t is actually reachable how could a  branch output no  um  [Music]  that must mean  that it does not guess  t as one of the reachable nodes because  it's going through all of the nodes here  um you know it's going to all the nodes  and it's picking them as reachable or  not  if it pick t is one of the reachable  ones then it's going to output yes  because it will find you know it'll  either output yes or if it doesn't find  the right doesn't guess the right path  it'll end up rejecting on that uh on  that path but some path will will end up  saying yes so if t is reachable  and  uh if you guess if you know t is  reachable and you guess t  you know you guess u is reachable one at  the point when u equals t you will end  up outputting s  the only way you could not  not  output yes is if you guess that node is  unreachable  but then your count is not going to add  up right  because you you wouldn't you did not  find all the reachable nodes if t is one  of the reachable nodes and you know  there are 100 reachable nodes  and you skipped over t as one of the  ones that you say is unreachable you at  best can only find 99 reachable nodes  and and you're not going to end up  saying no you're going to end up  projecting  so it's a very good question but you  have to think through what's going to  happen here that's  this c here is kind of a check  it's almost like you know um  uh  well in  it's like a checksum if you know what it  is it's so it makes sure that everything  that you if you got to see  if you got the if  if k equals c at this point that means  you actually found all the reachable  nodes so c is kind of a check  that you found all the reachable nodes  um  right so  if k equals c at this point you have  found every reachable node  and if t was one of the ones that are  reachable you found t  uh  okay let's see  um  is the reason we do this with c  essentially so that we know and we can  stop guessing and correctly identify if  it's impossible to reach t  well it's not a matter of it it's not a  matter of stopping guessing it's it's  it's a  it's a check that we found everything  because we're going to go through  and do all the guessing for every node  no matter what  so we're not going to stop anything  early unless we find that t is reachable  then then we can stop early but if to  show that t is not reachable we have to  go through the whole process  um  how can we intuitively see that we don't  have contradictory branches that's sort  of i was trying to say that just now i  don't know i i hope that got through  you can't have contradictory branches  because if you got to this stage here  you have found all the reachable  nodes so you've at this stage if you got  to six  you have made all correct guesses  you have found all the reachable nodes  you have convinced yourself that they're  all reachable um by  guessing the pairs to them and you've  checked that you have the right number  of reachable nodes because it equals c  so you must have found them all  so you cannot have a contradictory  answer because either t was one of the  ones you found in which case you would  have already said yes  or otherwise you found them all and t  was not one of them and so you're going  to say no you can't have both both  things cannot happen okay let's move on  so the next thing we're going to do is  the next slide is exactly the same as  this slide  except instead of saying  is um  is t reachable i want to know is it  reachable within d but within distance d  okay so um  which is going to mean exactly the same  procedure can i get  uh  instead of asking can i get  um  um  uh  from s to t  with a path of any length of course it's  going to be most length m now i want to  know can i get to it from s to t by a  path of most length d  these are number of edges in the path  say  um  and uh  that's the same procedure because  instead of i'm just going to cut things  off at  a d  but if i if i if i know in advance how  many nodes are reachable within d  um  i'm going to find all the nodes that are  reachable within d and c was t one of  the ones reachable within d  it's the same exact idea so here here is  the next slide which kind of shows that  uh so here here's the definitions  path sub d means reachable  by a path of length of most d  okay so um  r sub d is all of the ones that are  reachable  by a path of that length  and c sub d is the count it's a number  that are reachable uh within d  um so if you understood the last slide  hopefully this slide will seem kind of  obvious to you i'm gonna just highlight  all the changes  so if i can now calculate  c sub d which is the number reachable by  a path at most of length d  then i can test whether or not nodes are  reachable by a path without length  first i calculate c sub d  i go i pick every node as being  reachable within d or not  now i just have to check that my path  that i'm guessing has a length of most d  instead of length at most m which is  what i had before  keep a count of the ones that i found if  that count  equals c sub d then i know i found them  all if it's not equal to c sub d  then uh i've made some bad choice along  the way and i can just punt and say i  don't know  um  and if t was not one of the ones that  i've shown to be reachable within d  then i know it's not reachable within d  and so um  uh i can say no  okay so i don't know if this merits any  additional questions um but this is  really the same  it's just a repeat of the previous slide  what's kind of amazing  is now  the last slide  is  going to be again a repeat  uh let me just let me just force out of  where we're going but feel free to ask a  question on this or on the first slide  if you didn't on the previous slide if  you didn't get that um also we can try  to help you out with that  um  the next slide what i'm going to do  is show  how to compute  all these c values  and i should mention um  the value c which is the total number  reachable is going to be the same as c  sub m  reachable with an m the number of nodes  of the graph so if i can get up to c sub  m i'm done  um  and what i'm gonna show you is that  uh  knowing c sub i i can compute c sub i  plus one or c sub d i can compute c sub  d plus one  since i'm using d as my index here  basically  so c sub zero we know is just s well r  you know it's just one because you can  read just  start with s that's the only thing  reachable with zero  and then once i  know that i can fi figure out c sub 1 c  sub 2 c sub 3 and so on and then i get  the c sub n and then i have the count of  the total number reachable and then i  can test the path function  um  okay so the trick now is being able to  count  uh  given c sub d i would like to figure out  what is c sub d plus one  now how am i going to do that what i'm  going to do  is that's my goal what i'm going to do  is  something in between  i'm going to do a theorem just like this  but instead of  given c sub d instead of computing paths  of d i'm going to compute paths of d  plus 1.  so  knowing  how many are reachable from d  i'm going to give a test for whether  things are reachable within d plus one  and the fact is that's easy  because this thing already tells me how  to compute whether i'm reachable within  d  and being able to be reachable from  within d plus one means i have an edge  from something that's reachable within d  so if i can figure out which are  reachable within d  well and i just want to say do i have an  edge  you know  uh  do i have an edge from from one of the  nodes that are reachable within d then  i'm reachable within d plus one  then if i can test whether individual  nodes are reachable within d plus one i  can count  how many nodes are reachable within d  plus one that was that very first easy  theorem that i showed  so i know there's a lot of pieces here  that you have to put together but in the  end each p each individual piece is not  that bad  um  okay i don't know how many of you have  followed me oh no this is not supposed  to be here uh  there we go  so here is the last part  which again is just a simple  modification of what what the previous  slide had  so  i'm going to show how to compute  the path d plus 1 function so testing if  there's a path of length d plus 1 from s  to some node t  but only knowing how many nodes are  reachable within d  so i'm going to find all nodes that are  reachable within d  just like i did before but see if any  one of those nodes has an edge  to t  not necessarily that one is equal to t  because that says that t is reachable  within d but i want to know does it have  an edge to t that means t is reachable  within d plus one  so if i find all the nodes that are  reachable within d  and t turns out to be reachable from one  of those with it by an edge  then t is reachable within d uh d plus  one  and if d is not reachable from any of  those nodes with an edge then t is not  reachable when d plus one  i hope you're following me i'm not sure  you are uh  so anyway that's the that's the  algorithm here and  uh  uh the corollary is that you can compute  c sub d plus one from c sub d because if  you can count the path you if you can  test for each node if it's reachable as  i mentioned before you go through all  the nodes see whether the reachable and  d plus  d plus one and then  count them up  now i have c sub d plus one and now i'm  done  because  uh  uh  i'm gonna  compute each d plus one from the from  the value d that i previously computed  i'm going to do that from for all these  should say zero here actually um  and except  uh if the path says if this if the path  function now says there's no that the  answer is no  because i'm trying to do  you know the complement of the path  language  and reject of the path uh thing for m  says yes  and that's my non-deterministic  algorithm for the path  complement  uh problem  anyway  maybe you need to look at a little bit  offline um  it's presented in a little bit different  way in the book uh i don't know if that  will be more or less clear to you but i  kind of i think this has been a little  bit more unpacked for the purposes of  the lecture  um  so let's just see do we i'm not getting  any questions which probably means  uh  uh i've lost a huge chunk of you um but  uh the good news is we're gonna move on  to a different topic  so  um  but feel free to ask a question on this  if you want  or we're going to shift gears now to  talking about the hierarchy theorems  which is going to be the second half of  the lecture  also not so easy i have to say  probably a little less technical than  this one is but  it's also this are going to be  um spending time on just mainly just one  theorem  but anyway so looking ahead to what  where we're going and then we'll have a  break  um  we uh  well what we've shown so far  these are the major complexity classes  i'm not not including let's say the the  complementary classes the co-np uh type  classes um  these are the major complex classes  we've seen so far  and as we've seen they form a hierarchy  of containments  some of those containments trivial and  some slightly less trivial  but we have not shown whether any of  these classes are different  you know we've pointed out that there  are some unsolved problems here but do  we know any of these classes differ from  each other  or could it all collapse down to l  and the answer to that is  we do know that p space and l are  actually different  that we can prove  and it's a um  it relies on the theorem  that says if you give a  turing machine more space  then you can do more things  so because p space  is much is a bigger bound than log space  we know we can do more things  in fact  because  n l is contained within  log squared space deterministically and  p space is bigger than that we actually  can separate p space and nm  so we're going to prove that today  um so the basically the idea of the  theorem says that if you give a turing  machine  a bit more time or a bit more space then  it can do more  though you have there are some  conditions on that that we have to uh  we'll get into  one of the conclusions that we'll show  is that  um  time n squared if you compare with time  it's you know the time com the things  you can do in n squared time uh versus  the things you can do in cube time there  are more things you can do in n-cube  time then you can what you can do with  n-square time i mean that's what you  would expect  um but  it's not the case that everything we  expect in uh complexity theory we can  prove this is one of the things we can  prove  so as you add more time you can do more  things so this is a proper subset here  so there's there are some things in time  n cubed that are not in time n squared  and ditto for space  okay um  so that's going to be  that brings us to our  coffee break  and so feel free to shoot me any  questions about  what we've done so far or anything else  and otherwise  we will um  launch our timer and i'll see you in  five minutes  okay so i get getting some good  questions here could we also make a  solution this is getting back to that um  um  uh  um  the logs the nl equals coin l somebody's  saying could we just  um make another selection just by  non-deterministically choosing c  vertices  and then not and then checking that  they're all reachable  that's effectively what we're doing  but  be careful that we because we cannot  store c vertices  so that's why we're doing them one at a  time  we can't guess all c vertices up front  because where you're going to store all  that we only have log space  okay another so maybe somebody's asking  how much working space do we need for  storing the intermediate steps i'm not  sure what intermediate steps you mean  but if it's all the ci values  you know see  going from c0 to c1 to c2 to c3 we don't  store those  we all you need you need c sub d  to calculate c sub d plus one and then  you forget c sub d you couldn't store  all the c values but you don't need them  all you only need the most recent one to  go to the next one okay somebody's  asking can i go over why the complement  of path in nl  implies nl equal coin l  because the complement of path  is  uh essentially it's co n l complete i  mean l it is um  so  everything in nl is reducible to path  the  everything in cohen l  is reducible to the complement of path  by the same reduction  um and so if you can do the complement  of path in any uh class you can do all  of the complements of nl languages in  any class and so you can do the  complement of path in nl you can do all  the  connell problems in out in  nl and so then nl equals comma now  you have to you have to just think  through the logic of it this is not that  that part is not hard um  why it's enough to solve the path  complement problem in nl that does  everything else because it's a good so  the same completeness phenomenon that  we've been seeing  okay somebody's asking about the the two  sat problem that we talked about last  time um and is it um  and you know i i pointed out that that's  in  nl the you know the the two side problem  well the complement of the two set  problem  uh  you know the unsatisfiable two set  formulas that that's an nl language  because you can basically look for a  contradiction uh non-deterministically  in log space  um  that's  i think i probably won't be able to  explain that in a minute but maybe we'll  have our recitation instructors cover  that in recitation  um it's a nice proof not very hard but  it's it's an it's a nice proof  it's it's it's it's it's you know it's  not  it's something you have to do you have  to think about you you have to argue but  it's still it's not super hard um  understanding the two sad problem and  you know the complement of two set we  showed is in nl  and because nl equals coin l also the  two sat problem itself without  complementation is in nl  and in fact is nl complete  okay  um  i think we're going to have to move on  uh i'll stick around after lecture in  cases any questions that i can answer  quickly at that point  sorry if i couldn't get to your question  just now  all right  continuing on here  okay  shifting gears the space hierarchy  theorem  so  um  so as i mentioned i think um  uh maybe it's good to just go back  to this slide here um  we're going to do the time and space  hierarchy theorems which show that if  you can do a little bit more if you give  a little bit more time or a little bit  more space you can do more things  we're going to do the space  case  first because that actually tends to be  slightly  for certain technical reasons slightly  easier  so  uh  space hierarchy theorem so here is  the statement of the theorem and  it says for any bound  think of s is going to be some  you know  space bound  and again  f has to satisfy some technical  condition in yellow remember that it's  yellow because that's going to be  relevant later  um  uh so there's going to be some technical  condition  um  for no matter what function you you have  whatever space bound you have as long as  it satisfies this condition which is a  mild condition but you need it uh  whatever space bound you have um you can  find a language a  which requires exactly that much space  so if f is like n cubed we're going to  find a language a that requires n cubed  space if it's n to the hundredth we can  find the language that requires into the  100th space  and cannot be done within 1999 space  whatever it is  you can find a language that requires  exactly that much space and if you like  it a little bit more formally so that  means that it can be decided in that  much space  but it cannot be decided in less space  okay  framing it in a slightly different way  in terms of our space classes  um i'm going to define a notion which is  you know kind of it's not  it's  not said this way in the book but maybe  it's a helpful way to write it down  it's space little o of f of n  so those are all the things that you can  do by a function that's little o of f of  n  in space  so space little o of f n is properly  contained within space f of n  in other words there's something here  which is not in there  okay picture pictorially  i'm going to exhibit some language some  explicit language a  which i can do in this much space but  not in any less  now you know you can sort of think of  this as a little bit like the situation  for context-free languages and regular  languages where we exhibited a  particular language that differentiated  that was it context-free but not regular  and we're going to kind of do the same  thing now  but the one key difference is that in  the case of separating the context free  and the regular we could give a nice  language like 0 to the k1 to the k  here the language is not going to be so  nice to describe  it's going to be the language that some  turing machine we're going to give  decides  but you're not going to be able to get a  nice simple understanding of a it's  going to be whatever that turing machine  does and so in that sense it's not a  very natural language um that's easy to  sort of get your mind around um  so the i outline and really you don't  have to  worry about this but maybe it helps it's  really going to be a kind of a  diagonalization proof  um  the way this machine d  um is going to operate so d is going to  give you my language a  um  so d is going to be designed and i'm  going to show you d on the next slide  uh d is going to run within my target  space bound f of n  and here's the key here's the kicker  d is going to be designed to make sure  that its language cannot be done in less  space  and the way it does that  is it makes sure that its language is  different from any language that  that is um  decidable by a turing machine in in less  space and it's going to be different in  at least one place  so any  d is going to guarantee that its  language cannot be done in little o of f  of n space because it's going to be  different from every language that's  doable in little o of f of n space  somewhere  okay that's the point and then the  language a is going to be  the language of this  uh turing machine d  okay so it looks like a tall order the d  has to make sure that each  you know that that for every machine  its language differs  uh from that machine's language if that  machine is running in little of f event  space  but it's basically going to be a  diagonalization  so  for all of the different possible inputs  to d  that input is going to actually code up  a machine  uh on which  we're going to make sure that we're  different from that machine if it's a  small space machine  okay  let's see  so i can take a couple of questions here  does f have to be computable so that's  going to be one of the conditions um  that we're going to have to guarantee  where f satisfies the technical  condition yeah it's going to end up  being half it's going to be computable  but that's not enough  um  good question though  uh  okay so let's let's  let's move on from there okay now  so here is now what's what my job is to  give you this turing machine d  so d  these language is going to be my my  language a which i can which requires f  of n space cannot be done unless  okay oops i need to  i need the full slide here so i have to  take myself out  um  [Music]  all right uh now  this is my goal i want to exhibit this  language a which i can do in this much  space but not in any less  and  so i'm going to give this machine d as i  mentioned where a is d's language  d runs in order f event space and that  sort of that achieves this part  and d it makes sure that its language  cannot be done  in any less space  so that achieves this part  so it's different from the language of  any machine that runs in little o event  f of n space  okay um  [Music]  so uh  this is how d is gonna work  okay i'm gonna try to give you a little  picture uh to help help see the to  accompany the description  uh so d  gets its input w which is of length n  the very first thing d does  is it marks off  f of n's space  because it's only allowed to use  we're only going to allow d to use f of  n space because otherwise we're in  danger of  d  not  of a not being in in space f of n so d  is going to guarantee that by making  sure  it's going to mark off f of n space  and if it ever tries to use more than  that it just rejects  and but  by virtue of that we're sure that these  language is in space f of n because d is  an f of n space turing machine and it's  going to be the side  okay so  this part  so far is not too hard  okay now we're going to start getting  into the meat here  so  if w  um  now what we want to think of w as a  description of a machine that we're  going to feed that's that's going to run  on w so this is going to a little bit  you know  back to an earlier when we talked about  diagonalization so don't get  thrown off by this um  we if  we're going to think of w not only as  the input to d but it's also going to be  the description of a machine  and if it turns out that w is doesn't  describe anything it's just a jump w  then we're not not interested we're  gonna we're just gonna reject on that w  we're only interested in the w's that do  describe some machine m  okay  so if m  if w  uh describes some machine m  then we're gonna run m on w  and we're going to do the do the  opposite of what uh what m does  that's the whole idea  we're just going to uh make sure that  what we're doing is not the same as what  emma's doing  so at a high level  the the basic idea for this  is is not hard um  so we're going to simulate m on w  if m x rejects then we'll accept and if  m accepts then we'll reject we're just  going to do the opposite  um  and i think that is  so we have to be careful when we do the  simulation this is a little bit of a  detail but you know  this is a proof where you need to pay  attention to some details  um  the cost of simulating  m  on d  is only a constant factor  um because if m uses a certain amount of  space  when d is simulating m you know m may  have a larger tape alphabet than d does  but these can then encode um m's tape by  using several cells for each of m's  cells but it's only going to be a  constant factor and that's important  here because  um we have to make sure that  you know if this was a big blow up um d  would not be able to run m  um  i i think i'm sort of arguing the  details without making sure we  understand the fundamental concept um so  let me back up  the point is that d is doing something  the opposite of  m now  uh d can't be different from every m  because d itself is a turing machine of  course but but the thing is is that  d is only running within f of n tape  cells so it has to be able to do that  simulation of m within that amount of  tape if m is using a lot of tape then d  is going to use a lot of tape and it's  just going to reject  so this is only going to really come  into play  getting being able to simulate m if m is  using a small amount of using a small  amount of space so that d can do the  simulation  okay so let's just see maybe  uh  so they're going to be some issues here  but before i get to that let's just see  what uh what your questions are  how can a turing machine know if w is  encoding some other turing machine  no that's simple  you know what what is a coding of a  turing machine it's just you know the  standard we have a standard coding um  it's just you know coding the rules of  the machine so it has to have states  transition function blah blah blah so it  just has to be some you know whatever  our encoding for the turing machine is  we can always test whether  a string is a legitimate encoding of a  turing machine so that that shouldn't be  bad um  somebody says why do we reject if we use  more than f n cells isn't it okay to use  order f of n  yes it could be but  we have to cut it off somewhere  you know  it might be it's okay we could use two f  of n we could use 10 f of n but we have  to have some constant for d  and let's just kind of simply constant  one so d has to run within f of n cells  um and that's going to guaran that's  going to be good enough for us  okay do we have to make sure that m runs  a little of f of n so we can't really  tell  whether m is running in little o of f of  n  or we can tell us whether we can finish  the simulation  so that's actually going to be  maybe you can just hold off on that  question because there is a point that  we have to follow up on in that which is  um  uh just because  you know we may or may not be able to  finish simulating m  on this w  doesn't necessarily tell us what the  asymptotic behavior of m is but we'll  have to look at that in a bit okay so  somebody's saying what happens if m  loops on w that's going to be one of our  issues we have to deal with that's a  good question there step two alone can  use them more than f of n cells  yeah step two alone can use more than f  of n cells if it does we're just going  to end up projecting  okay so we're getting good questions  here some of them we're going to which  i'm going to address anyway so why don't  we just move on um  okay  so here is sort of a question i think  this is one of the questions that that  related to one of the ones that got  asked what happens if it runs in little  of f of n space so we we remember what  we're trying to do is be different from  every small space you know little o of f  of n space machine  so what if m runs a little o of f of n  space but has a big constant so what i  mean by that concretely is suppose d is  an n cubed space so suppose we're trying  to  get a  in in cubed space  but show what's not in n squared space  d is going to run an n cubed  and what and we have to make sure that  any machine that's running in n squared  space cannot do the same language  um so we're going to be different from  that but the problem is that uh in  uh  the machine m  might be running in  n squared space but with a huge constant  so it might be running in a million n  squared so that's still a machine that's  running in  a little o  of n cubed and we have to be different  from it but for the particular w we're  working on we might not have enough  space to run m because of the huge  constant it that con  the asymptotic behavior is only going to  be relevant well for large w for smaller  b we may not see that we may not have  enough space to run m so  um  what are we going to do to fix that  we're going to run that m on infinitely  many different w so it's going to be  infinitely many different w's  that are all going to encode the same m  and the way i'm going to do that  is by  uh  thinking of w as representing m  but having  an unbounded number of trailing zeros  after that  so i'm going to strip off the very first  thing i'm going to do with w is i'm  going to strip off the trailing zeros up  into the final one i'm going to remove  those  and then take the rest  and as the description of the machine  so now i'm going to have potentially  w's that have an enormous number of  zeros at the end  big enough so that i can see the  asymptotic behavior of m  and that if m is really running in  little o of f of n space i'll have  enough space to run m to completion on w  and so then i'll be able to be different  from it  okay  um  so i'm kind of showing that over here so  here's a very large w  i'm going to strip off the trailing  zeros  the rest of it is just going to be m and  i'm going to run this m on the whole w  the entire w without the zero stripped  off so now m is going to be running on a  very large input  um  big enough so that d uh d which has  asymptotically more space than m does  will have enough space to run empty  completion  um  now another question that got asked what  happens if m loops  that's going to be a problem because d  always has to hold  and if it just blindly simulates m  then d might be looping on m  none of m is going to use a lot of space  by the way  because then d is going to catch it in  step one but if m uses  his loops on a small amount of space  then uh d  might end up looping as  presently constructed so what i'm going  to do  is i'm going to put a counter which  makes it stop if it runs for  2 to the f of n space so basically  because that's how long d could possibly  run without looping anyway  m could be running without looping  anyway and so we're going to run it for  this amount of  this number of steps and  uh i'm going to reject if it hasn't yet  halted as well as uh that because it has  to be looping at that point anyway um  and so it's not interesting for us  it doesn't matter what we're going to do  if it hasn't halted because  m is not a decider  and the last thing is how to compute f  um  so i'll try to address some questions  here in in our remaining time  uh how to compute f  so to mark off f of n cells we also have  to compute f i didn't think anybody any  of you guys asked that question except  maybe sort of the very beginning about f  being a computable function  certainly f is going to have to be  computable but not only does it have to  be computable it has to be computable  within the space bound  and that's just going to be a condition  we're going to impose on f it's so  called space constructable namely that  you can compute it within its own space  bound  and all nice functions that we care  about are going to be space  constructable  so  it's not doesn't turn out to be  an obstacle to applying  uh the hierarchy theorem but it is a  condition that we need it actually is  not true without that condition  um  okay let's let's just oh this i have a  check in here maybe we can take a couple  of questions first  some of you are anticipating my check-in  actually which is good um  so let me hold off on those  sorry a bit confused about what is m  can we say d as input m and simulate m  on yeah so  uh somebody's saying  can we say that d has input m  and simulates m on itself yes that's  exactly what's happening the reason why  we're doing that is because we have to  cover all possible m's  so as we get all possible inputs w  they're going to range over all possible  ends and so every possible m is going to  get addressed to see if we can run it um  within the space bound and be different  from it  d's job is to be different from each of  those ends  but it's not you know again there were  some details here that got raised in  these issues um  but in a sense this is just kind of  more technical  i would focus on understanding what i  originally wrote down because that's the  main idea the rest of it is just kind of  implementation details  okay so why don't i  can i give an example of a non-space  constructable function  yes  log log log n space you cannot compute  log log log n space within log log log n  space  and in fact it's known  that  there's nothing new  between constant space which is just  regular and log log log n space anything  you can do in log log log n space is a  regular language  so the the hierarchy theorem doesn't  doesn't won't apply there because well  it applies but that's not a it doesn't  well you know it's not it's not space  constructible  to find  higher level um you know large  um  non-space constructable functions you  can do it but they're  you know you you  they're not easy to describe  okay let's do our check in here  what happens when we run dion itself i  got a couple of people asking me about  that so that's a this is just a good  lead into our check-in and this a little  you really have to understand what's  going on  to see what does d do  when if you feed in itself maybe with  some trailing zeros because remember the  algorithm strips off trailing zero so  what is what does it do in that case um  so here are my  options there you can get to pick which  ones which one  you uh you think is the answer so i'll  give you another 30 seconds on this  because this requires a little bit of  thinking  if you want to invest in it all right  i'm wrapping this up guys  five seconds to go  okay i'm gonna end it so get your  participation points in  a bunch of you have not uh said anything  come on  um i can see the count here and it's  there's a  three or four of you are not not  answered well you're going to lose out  closing  all right so the right answer is in fact  c it does reject let's just understand  what happened is definitely we don't get  a contradiction i mean this is an  algorithm i just described it's going to  do something  um  uh  i'm assuming the people who picked e are  having fun as i did when i came up with  the check-in but um  uh not a question answer a doesn't is  not going to be good either because d  has to be a decider so it can't loop on  anything so the only sort of reasonable  answers are accept or reject  when you run d on itself  uh  what's it going to try to do  it's going to the very first thing it's  going to you know mark off f of n tape  cells  and then it's going to get its input  which is itself  tries to simulate itself on the same  input  that uh simulated d is also going to try  to mark off f of n tape cells  but um due to some simulation you know  there's going to be some cost to doing  the stimulation when the simulated d is  going to try to mark off f of n tape  cells it's going to blow  the original d  space bound and exceed the bound and so  d is going to reject up right in step  one  when it tries to get an input of its of  itself  so that's um  it's very clear what's gonna happen in  this it's just gonna reject  because of the for this reject this  reject in particular  and notice this would you know yeah  you know  okay let me not try to confuse it um  okay so that's that's that's all i want  to say about this um let's now move in  our remaining seven minutes to the time  hierarchy theorem which is very has the  same proof um but some of the technical  details are slightly different  okay um  so now if i give you a time bound again  we you have gonna have a face with the  same notion that you have to be able to  compute f within f's amount of time  so it has to be a time constructable i'm  not gonna define that um  so there's a language a which requires  that much time  um so it has to be decidable within that  much time but there's a slight  difference here and there's and this is  an artifact of the proof of the theorem  not because it's an absolute truth as  far as we know  uh it's not that it's not decidable  little o of f n you actually you can  only prove something slightly weaker  when you have one tape turing machines  that it's little decidable little o over  there's there's a slight um  uh  gap in what you can prove so it's not  only that you can't prove you it  requires little o but little o of  little of f of n over log f of n  is uh what you can prove that the you  get from this uh time hierarchy theorem  but let's not get caught up on that for  now um  uh  okay  so the proof outline is the same outline  as we had before um  uh  we're going to give a d that runs in  order f of n time  so it's  ensures that the language is in that uh  time complexity class time f of n  and it makes sure it's different from  every machine that runs faster  by some significant love by a log factor  faster  okay and so um why don't i  uh  show how that goes  um  the proof is in some ways  almost exactly the same  um  i'm going to give a d  which runs this much time  and it shows it's different from every  m that runs in  a lot less time  here is the algorithm for d  now it computes f n but it does  something a little different with f of n  remember in the space  hierarchy theorem  we marked off f of n space  now this f event is going to be used for  a different purpose it's going to be a  clock  and you have to shut m down  if it runs for more than f of n steps  not if it uses more than f of n space  because we're only interested in m's  that use  significantly less  than f of n time  so we're going to run an m you know for  f for some number of steps  whatever m says we're going to do the  opposite and only if we can actually  finish that simulation we'll be be able  to be sure that we're different from  what m is doing  um so this is the whole algorithm here  we don't have to do any further  modifications  and where's that login factor coming  from  it's actually coming from a funny place  um  and and you know you have to get into  the little bit of the guts of this  when you're simulating m on w  remember that m itself was described by  w  so you're going to have to write down a  copy of m  which is you know just  as described by w  and then so you're gonna and then you're  gonna have the tape that am is working  on  which is starting out with w on it  um  and uh  you have to be now you have to be a  little careful how you manage that  because if your description of m is just  sitting at the beginning of the tape  as you're simulating m  every time you do one step and modifying  the tape  you don't want to have to go back  to the beginning of the tape to look up  the next step of m  so you actually have to carry m along  with you  as you're doing the simulation  and you can do that by expanding the  tape alphabet of the tape  so that you can effectively have two  symbols  on  one cell one is going to be  for describing m and the other one is  going to be for just the for m's  for the simulation tape  and you'll be carrying m along with you  uh wherever your head is  so you don't have to go very far to look  up m  and  so that's all possible because that's  going to add only a constant factor  because m is fixed in size doesn't  depend on  you know  you know  for large inputs to m m is fixed  but the tricky thing here is  the counter to make sure we're not using  too much time  um  [Music]  the counter has size log f of n because  that's how big it has to count up to so  you should you can shut it down if it's  going to exceed the f of n steps  um  and  because you know you have to  run for a certain amount of time  and  carrying the keeping the counter nearby  has the counter now could be pretty big  and so that's going to cost you a log  factor of um  of simulation cost  to  move that counter around all the time  and so  that's why you have to  run for only a log factor less  so that you can actually finish within f  of n time as you're as you're required  to do  okay i realize that that's a mouthful  there and you may not have all  understood that  it doesn't matter it's not that critical  i think what i'm really more concerned  is you understand the the main idea of  the um the hierarchy theorem some of  these implementation details  you know you  if you don't get them  i wouldn't worry about it i i feel i  have to include them for completeness  sake and to be honest with you about the  proof but if you didn't follow  everything that's okay i do want to  understand the main idea though of the  algorithm  making sure that what it's doing is  different from what every machine is  doing if that machine runs in little o  of f of n space or or a small amount of  time you know little o of f n over log f  of n time  okay so and i think we're gonna uh  we're gonna end here so come pretty much  out of time uh i'm gonna stick around  for a little bit  um  in case there's any questions here oh  wait there's one let's check in though  um  let's let's look at this this is kind of  an interesting sort of follow-on to the  hierarchy theorem um if you look at the  two questions does l equal p and does p  equal p space  these are both unsolved problems  does the what if anything does the  hierarchy theorem tell us about those  questions  and it's kind of interesting that there  actually does well i'll leave it to you  to  uh  tell me if you can see what what it  might actually be telling you closing  get your answer in  okay one two three  i feel like i'm running an auction house  here  i should have a gavel  okay  yes in fact we know that you know these  are separated so it's not even though we  don't know if l equals p or p equals p  space they can't both be equal  because then um l would equal p space  and we know that's false  um  so at least one of these has the answer  no  okay so with that let's wrap up today's  lecture  you know we prove these hierarchy  theorems and why don't we just uh  um  i'm going to  uh  shut us down here but but before well  we're over so you can feel free to go um  but i'll stick around in case anybody's  any questions for a few minutes anyway  um and then uh  we'll call it a day  since we just showed space and is a  proper subset of space n to the k for  any k why can't we also say  space n is a proper subset of p space  yes space n is a proper subset of p  space  yeah so somebody just asked we just  showed that space n is a proper subset  of space n to the  k does that also say that space n is a  proper subset of p space definitely  any  um  space n to the k is a proper subset of  space n to the k plus one  which is a subset of p space so any  fixed polynomial is going to be a subset  of p space because p space includes all  the polynomials  which of the two unsolved problems  uh whoops  do i think is more likely likely to be  true  well  i think most  i mean i would bet that both of these  are not equal so both of these have  answered no  um  uh it would be weird you know  i mean you think l equals p  that anything you can do in polynomial  time you end with log space log space is  incredibly weak  and and p space is incredibly strong  um  i would be shocked if either of these  were equal  so we just  the problem is that we don't have a  method  for proving  um  problems  are are actually  have high complexity of of any sort we  don't know how to show things outside of  l don't know how to show things  outside of p  except by using the hierarchy theorem  diagonalization is the only method that  we have for showing things or outside of  classes  and there's reason to believe that as  we'll this will get to i think next  lecture in fact  there's kind of reasons to believe that  the hierarchy theorem type argument  which is diagonalization is not going to  answer those kinds of questions um so we  need a different method and  diagonalization is all we got  good question though  if i didn't get to answer your question  you have a question for me ask it again  because it's got buried  so it means we are very far from  disproving p versus np is that right  it could happen tomorrow  you know how do you how do you how can  you tell it doesn't you know it it seems  clear  that the present state of mathematics as  of right now  is  you know we don't have a clue  how to answer those kinds of questions  and it's not obvious that we've even  made any progress  um but  you know  that's the nature of the game that's the  nature of the beast  you know somebody gets a good idea and  all of a sudden lots of things uh can  change  and that can happen and then in any  point maybe one of you guys  when were these results first  this is the the stuff okay the hierarchy  theorem is old  that goes back to the very when time  when time classes were first defined i  think is one of the first results to  show the hierarchy  and  that's  late 60s  the  nl equal coen l i think i mentioned was  like mid 1980s  much later  i mean from your points of view it was a  bit it was back in the  cave  cave age uh either way but uh yeah but  but the the hierarchy team that that's  actually pre predates uh  my coming into the field but but the but  the nl equal conel that was something  that i was  i personally experienced how surprised  people were  okay i think i'm gonna  send you off all off on your way good  good having you here and i have a good  weekend everybody  and um  i will see you on tuesday  bye  you
 

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00:07:50,230 --> 00:07:52,950

 

253
00:07:52,950 --> 00:07:55,749

 

254
00:07:55,749 --> 00:07:58,710

 

255
00:07:58,710 --> 00:08:00,070

 

256
00:08:00,070 --> 00:08:03,830

 

257
00:08:03,830 --> 00:08:05,350

 

258
00:08:05,350 --> 00:08:09,589

 

259
00:08:09,589 --> 00:08:11,350

 

260
00:08:11,350 --> 00:08:12,950

 

261
00:08:12,950 --> 00:08:17,189

 

262
00:08:17,189 --> 00:08:18,469

 

263
00:08:18,469 --> 00:08:19,830

 

264
00:08:19,830 --> 00:08:19,840

 

265
00:08:19,840 --> 00:08:21,189


266
00:08:21,189 --> 00:08:22,570

 

267
00:08:22,570 --> 00:08:22,580

 

268
00:08:22,580 --> 00:08:23,749


269
00:08:23,749 --> 00:08:25,670

 

270
00:08:25,670 --> 00:08:25,680

 

271
00:08:25,680 --> 00:08:26,550


272
00:08:26,550 --> 00:08:27,909

 

273
00:08:27,909 --> 00:08:30,309

 

274
00:08:30,309 --> 00:08:31,749

 

275
00:08:31,749 --> 00:08:33,670

 

276
00:08:33,670 --> 00:08:37,110

 

277
00:08:37,110 --> 00:08:39,750

 

278
00:08:39,750 --> 00:08:41,269

 

279
00:08:41,269 --> 00:08:43,589

 

280
00:08:43,589 --> 00:08:45,509

 

281
00:08:45,509 --> 00:08:47,190

 

282
00:08:47,190 --> 00:08:48,949

 

283
00:08:48,949 --> 00:08:50,949

 

284
00:08:50,949 --> 00:08:53,670

 

285
00:08:53,670 --> 00:08:55,750

 

286
00:08:55,750 --> 00:08:55,760

 

287
00:08:55,760 --> 00:08:56,870


288
00:08:56,870 --> 00:08:59,750

 

289
00:08:59,750 --> 00:09:01,990

 

290
00:09:01,990 --> 00:09:03,430

 

291
00:09:03,430 --> 00:09:06,949

 

292
00:09:06,949 --> 00:09:09,910

 

293
00:09:09,910 --> 00:09:13,190

 

294
00:09:13,190 --> 00:09:14,949

 

295
00:09:14,949 --> 00:09:14,959

 

296
00:09:14,959 --> 00:09:16,150


297
00:09:16,150 --> 00:09:19,430

 

298
00:09:19,430 --> 00:09:21,110

 

299
00:09:21,110 --> 00:09:22,870

 

300
00:09:22,870 --> 00:09:25,750

 

301
00:09:25,750 --> 00:09:27,670

 

302
00:09:27,670 --> 00:09:30,150

 

303
00:09:30,150 --> 00:09:30,160

 

304
00:09:30,160 --> 00:09:31,269


305
00:09:31,269 --> 00:09:32,630

 

306
00:09:32,630 --> 00:09:34,150

 

307
00:09:34,150 --> 00:09:36,630

 

308
00:09:36,630 --> 00:09:38,230

 

309
00:09:38,230 --> 00:09:39,350

 

310
00:09:39,350 --> 00:09:41,910

 

311
00:09:41,910 --> 00:09:43,670

 

312
00:09:43,670 --> 00:09:45,350

 

313
00:09:45,350 --> 00:09:46,790

 

314
00:09:46,790 --> 00:09:48,150

 

315
00:09:48,150 --> 00:09:49,269

 

316
00:09:49,269 --> 00:09:51,269

 

317
00:09:51,269 --> 00:09:53,350

 

318
00:09:53,350 --> 00:09:56,550

 

319
00:09:56,550 --> 00:09:59,670

 

320
00:09:59,670 --> 00:10:01,590

 

321
00:10:01,590 --> 00:10:03,590

 

322
00:10:03,590 --> 00:10:05,829

 

323
00:10:05,829 --> 00:10:05,839

 

324
00:10:05,839 --> 00:10:07,350


325
00:10:07,350 --> 00:10:08,870

 

326
00:10:08,870 --> 00:10:10,630

 

327
00:10:10,630 --> 00:10:11,990

 

328
00:10:11,990 --> 00:10:14,310

 

329
00:10:14,310 --> 00:10:14,320

 

330
00:10:14,320 --> 00:10:15,750


331
00:10:15,750 --> 00:10:19,190

 

332
00:10:19,190 --> 00:10:23,350

 

333
00:10:23,350 --> 00:10:23,360

 

334
00:10:23,360 --> 00:10:24,230


335
00:10:24,230 --> 00:10:25,910

 

336
00:10:25,910 --> 00:10:28,870

 

337
00:10:28,870 --> 00:10:30,550

 

338
00:10:30,550 --> 00:10:33,110

 

339
00:10:33,110 --> 00:10:35,590

 

340
00:10:35,590 --> 00:10:37,509

 

341
00:10:37,509 --> 00:10:40,630

 

342
00:10:40,630 --> 00:10:42,630

 

343
00:10:42,630 --> 00:10:45,990

 

344
00:10:45,990 --> 00:10:46,000

 

345
00:10:46,000 --> 00:10:47,110


346
00:10:47,110 --> 00:10:48,310

 

347
00:10:48,310 --> 00:10:49,990

 

348
00:10:49,990 --> 00:10:51,990

 

349
00:10:51,990 --> 00:10:53,990

 

350
00:10:53,990 --> 00:10:56,790

 

351
00:10:56,790 --> 00:10:59,590

 

352
00:10:59,590 --> 00:11:01,269

 

353
00:11:01,269 --> 00:11:03,190

 

354
00:11:03,190 --> 00:11:05,750

 

355
00:11:05,750 --> 00:11:07,990

 

356
00:11:07,990 --> 00:11:10,150

 

357
00:11:10,150 --> 00:11:12,710

 

358
00:11:12,710 --> 00:11:12,720

 

359
00:11:12,720 --> 00:11:13,829


360
00:11:13,829 --> 00:11:21,670

 

361
00:11:21,670 --> 00:11:23,110

 

362
00:11:23,110 --> 00:11:26,829

 

363
00:11:26,829 --> 00:11:28,949

 

364
00:11:28,949 --> 00:11:31,750

 

365
00:11:31,750 --> 00:11:34,150

 

366
00:11:34,150 --> 00:11:34,160

 

367
00:11:34,160 --> 00:11:38,230


368
00:11:38,230 --> 00:11:40,829

 

369
00:11:40,829 --> 00:11:43,350

 

370
00:11:43,350 --> 00:11:43,360

 

371
00:11:43,360 --> 00:11:44,389


372
00:11:44,389 --> 00:11:44,399

 

373
00:11:44,399 --> 00:11:45,829


374
00:11:45,829 --> 00:11:47,509

 

375
00:11:47,509 --> 00:11:49,190

 

376
00:11:49,190 --> 00:11:51,590

 

377
00:11:51,590 --> 00:11:51,600

 

378
00:11:51,600 --> 00:11:52,550


379
00:11:52,550 --> 00:11:52,560

 

380
00:11:52,560 --> 00:11:54,310


381
00:11:54,310 --> 00:11:57,190

 

382
00:11:57,190 --> 00:11:59,110

 

383
00:11:59,110 --> 00:11:59,120

 

384
00:11:59,120 --> 00:12:00,470


385
00:12:00,470 --> 00:12:06,069

 

386
00:12:06,069 --> 00:12:06,079

 

387
00:12:06,079 --> 00:12:07,590


388
00:12:07,590 --> 00:12:08,829

 

389
00:12:08,829 --> 00:12:11,190

 

390
00:12:11,190 --> 00:12:13,350

 

391
00:12:13,350 --> 00:12:15,350

 

392
00:12:15,350 --> 00:12:15,360

 

393
00:12:15,360 --> 00:12:16,389


394
00:12:16,389 --> 00:12:22,389

 

395
00:12:22,389 --> 00:12:24,389

 

396
00:12:24,389 --> 00:12:27,030

 

397
00:12:27,030 --> 00:12:29,509

 

398
00:12:29,509 --> 00:12:31,350

 

399
00:12:31,350 --> 00:12:34,069

 

400
00:12:34,069 --> 00:12:36,710

 

401
00:12:36,710 --> 00:12:39,430

 

402
00:12:39,430 --> 00:12:40,949

 

403
00:12:40,949 --> 00:12:42,870

 

404
00:12:42,870 --> 00:12:45,590

 

405
00:12:45,590 --> 00:12:45,600

 

406
00:12:45,600 --> 00:12:47,590


407
00:12:47,590 --> 00:12:50,710

 

408
00:12:50,710 --> 00:12:52,790

 

409
00:12:52,790 --> 00:12:54,389

 

410
00:12:54,389 --> 00:12:56,310

 

411
00:12:56,310 --> 00:12:57,990

 

412
00:12:57,990 --> 00:12:59,590

 

413
00:12:59,590 --> 00:13:01,509

 

414
00:13:01,509 --> 00:13:03,750

 

415
00:13:03,750 --> 00:13:06,230

 

416
00:13:06,230 --> 00:13:08,069

 

417
00:13:08,069 --> 00:13:10,230

 

418
00:13:10,230 --> 00:13:11,430

 

419
00:13:11,430 --> 00:13:13,350

 

420
00:13:13,350 --> 00:13:13,360

 

421
00:13:13,360 --> 00:13:15,910


422
00:13:15,910 --> 00:13:17,509

 

423
00:13:17,509 --> 00:13:18,829

 

424
00:13:18,829 --> 00:13:20,629

 

425
00:13:20,629 --> 00:13:21,910

 

426
00:13:21,910 --> 00:13:21,920

 

427
00:13:21,920 --> 00:13:23,110


428
00:13:23,110 --> 00:13:25,509

 

429
00:13:25,509 --> 00:13:27,030

 

430
00:13:27,030 --> 00:13:28,389

 

431
00:13:28,389 --> 00:13:28,399

 

432
00:13:28,399 --> 00:13:30,550


433
00:13:30,550 --> 00:13:32,389

 

434
00:13:32,389 --> 00:13:34,790

 

435
00:13:34,790 --> 00:13:34,800

 

436
00:13:34,800 --> 00:13:36,470


437
00:13:36,470 --> 00:13:38,550

 

438
00:13:38,550 --> 00:13:39,990

 

439
00:13:39,990 --> 00:13:41,110

 

440
00:13:41,110 --> 00:13:45,670

 

441
00:13:45,670 --> 00:13:48,230

 

442
00:13:48,230 --> 00:13:49,350

 

443
00:13:49,350 --> 00:13:50,790

 

444
00:13:50,790 --> 00:13:53,030

 

445
00:13:53,030 --> 00:13:54,470

 

446
00:13:54,470 --> 00:13:56,629

 

447
00:13:56,629 --> 00:13:58,870

 

448
00:13:58,870 --> 00:14:01,189

 

449
00:14:01,189 --> 00:14:04,150

 

450
00:14:04,150 --> 00:14:07,590

 

451
00:14:07,590 --> 00:14:09,110

 

452
00:14:09,110 --> 00:14:09,120

 

453
00:14:09,120 --> 00:14:10,230


454
00:14:10,230 --> 00:14:12,230

 

455
00:14:12,230 --> 00:14:14,470

 

456
00:14:14,470 --> 00:14:17,430

 

457
00:14:17,430 --> 00:14:21,670

 

458
00:14:21,670 --> 00:14:23,110

 

459
00:14:23,110 --> 00:14:26,230

 

460
00:14:26,230 --> 00:14:28,230

 

461
00:14:28,230 --> 00:14:30,949

 

462
00:14:30,949 --> 00:14:32,870

 

463
00:14:32,870 --> 00:14:34,710

 

464
00:14:34,710 --> 00:14:35,990

 

465
00:14:35,990 --> 00:14:38,230

 

466
00:14:38,230 --> 00:14:38,240

 

467
00:14:38,240 --> 00:14:40,470


468
00:14:40,470 --> 00:14:42,629

 

469
00:14:42,629 --> 00:14:44,069

 

470
00:14:44,069 --> 00:14:45,750

 

471
00:14:45,750 --> 00:14:47,750

 

472
00:14:47,750 --> 00:14:49,269

 

473
00:14:49,269 --> 00:14:49,279

 

474
00:14:49,279 --> 00:14:50,310


475
00:14:50,310 --> 00:14:51,910

 

476
00:14:51,910 --> 00:14:54,069

 

477
00:14:54,069 --> 00:14:57,509

 

478
00:14:57,509 --> 00:14:59,990

 

479
00:14:59,990 --> 00:15:02,230

 

480
00:15:02,230 --> 00:15:05,269

 

481
00:15:05,269 --> 00:15:07,990

 

482
00:15:07,990 --> 00:15:09,030

 

483
00:15:09,030 --> 00:15:11,670

 

484
00:15:11,670 --> 00:15:15,189

 

485
00:15:15,189 --> 00:15:17,430

 

486
00:15:17,430 --> 00:15:20,550

 

487
00:15:20,550 --> 00:15:22,310

 

488
00:15:22,310 --> 00:15:24,949

 

489
00:15:24,949 --> 00:15:26,470

 

490
00:15:26,470 --> 00:15:28,550

 

491
00:15:28,550 --> 00:15:32,230

 

492
00:15:32,230 --> 00:15:34,389

 

493
00:15:34,389 --> 00:15:35,910

 

494
00:15:35,910 --> 00:15:40,310

 

495
00:15:40,310 --> 00:15:42,629

 

496
00:15:42,629 --> 00:15:44,470

 

497
00:15:44,470 --> 00:15:44,480

 

498
00:15:44,480 --> 00:15:45,430


499
00:15:45,430 --> 00:15:48,310

 

500
00:15:48,310 --> 00:15:51,509

 

501
00:15:51,509 --> 00:15:53,269

 

502
00:15:53,269 --> 00:15:56,550

 

503
00:15:56,550 --> 00:16:00,949

 

504
00:16:00,949 --> 00:16:03,990

 

505
00:16:03,990 --> 00:16:05,990

 

506
00:16:05,990 --> 00:16:07,829

 

507
00:16:07,829 --> 00:16:07,839

 

508
00:16:07,839 --> 00:16:08,790


509
00:16:08,790 --> 00:16:08,800

 

510
00:16:08,800 --> 00:16:10,310


511
00:16:10,310 --> 00:16:10,320

 

512
00:16:10,320 --> 00:16:13,670


513
00:16:13,670 --> 00:16:15,829

 

514
00:16:15,829 --> 00:16:18,310

 

515
00:16:18,310 --> 00:16:18,320

 

516
00:16:18,320 --> 00:16:20,870


517
00:16:20,870 --> 00:16:23,110

 

518
00:16:23,110 --> 00:16:25,509

 

519
00:16:25,509 --> 00:16:27,590

 

520
00:16:27,590 --> 00:16:30,629

 

521
00:16:30,629 --> 00:16:34,310

 

522
00:16:34,310 --> 00:16:35,430

 

523
00:16:35,430 --> 00:16:37,269

 

524
00:16:37,269 --> 00:16:41,910

 

525
00:16:41,910 --> 00:16:45,749

 

526
00:16:45,749 --> 00:16:48,150

 

527
00:16:48,150 --> 00:16:51,749

 

528
00:16:51,749 --> 00:16:54,310

 

529
00:16:54,310 --> 00:16:57,509

 

530
00:16:57,509 --> 00:16:59,749

 

531
00:16:59,749 --> 00:17:01,509

 

532
00:17:01,509 --> 00:17:03,749

 

533
00:17:03,749 --> 00:17:06,230

 

534
00:17:06,230 --> 00:17:08,069

 

535
00:17:08,069 --> 00:17:10,230

 

536
00:17:10,230 --> 00:17:12,110

 

537
00:17:12,110 --> 00:17:13,990

 

538
00:17:13,990 --> 00:17:16,069

 

539
00:17:16,069 --> 00:17:17,429

 

540
00:17:17,429 --> 00:17:19,429

 

541
00:17:19,429 --> 00:17:21,510

 

542
00:17:21,510 --> 00:17:21,520

 

543
00:17:21,520 --> 00:17:22,549


544
00:17:22,549 --> 00:17:26,549

 

545
00:17:26,549 --> 00:17:29,590

 

546
00:17:29,590 --> 00:17:31,430

 

547
00:17:31,430 --> 00:17:33,350

 

548
00:17:33,350 --> 00:17:34,630

 

549
00:17:34,630 --> 00:17:36,789

 

550
00:17:36,789 --> 00:17:39,669

 

551
00:17:39,669 --> 00:17:41,029

 

552
00:17:41,029 --> 00:17:44,150

 

553
00:17:44,150 --> 00:17:46,710

 

554
00:17:46,710 --> 00:17:46,720

 

555
00:17:46,720 --> 00:17:52,150


556
00:17:52,150 --> 00:17:55,110

 

557
00:17:55,110 --> 00:17:56,310

 

558
00:17:56,310 --> 00:17:56,320

 

559
00:17:56,320 --> 00:17:57,909


560
00:17:57,909 --> 00:18:01,190

 

561
00:18:01,190 --> 00:18:02,549

 

562
00:18:02,549 --> 00:18:04,549

 

563
00:18:04,549 --> 00:18:06,549

 

564
00:18:06,549 --> 00:18:09,029

 

565
00:18:09,029 --> 00:18:09,039

 

566
00:18:09,039 --> 00:18:10,230


567
00:18:10,230 --> 00:18:12,950

 

568
00:18:12,950 --> 00:18:15,590

 

569
00:18:15,590 --> 00:18:17,590

 

570
00:18:17,590 --> 00:18:20,150

 

571
00:18:20,150 --> 00:18:23,029

 

572
00:18:23,029 --> 00:18:27,750

 

573
00:18:27,750 --> 00:18:27,760

 

574
00:18:27,760 --> 00:18:28,549


575
00:18:28,549 --> 00:18:32,789

 

576
00:18:32,789 --> 00:18:34,549

 

577
00:18:34,549 --> 00:18:36,710

 

578
00:18:36,710 --> 00:18:39,830

 

579
00:18:39,830 --> 00:18:41,750

 

580
00:18:41,750 --> 00:18:41,760

 

581
00:18:41,760 --> 00:18:44,390


582
00:18:44,390 --> 00:18:46,710

 

583
00:18:46,710 --> 00:18:46,720

 

584
00:18:46,720 --> 00:18:47,590


585
00:18:47,590 --> 00:18:49,990

 

586
00:18:49,990 --> 00:18:52,390

 

587
00:18:52,390 --> 00:18:55,669

 

588
00:18:55,669 --> 00:18:56,870

 

589
00:18:56,870 --> 00:18:58,789

 

590
00:18:58,789 --> 00:19:00,630

 

591
00:19:00,630 --> 00:19:02,630

 

592
00:19:02,630 --> 00:19:05,029

 

593
00:19:05,029 --> 00:19:06,549

 

594
00:19:06,549 --> 00:19:10,070

 

595
00:19:10,070 --> 00:19:12,390

 

596
00:19:12,390 --> 00:19:14,549

 

597
00:19:14,549 --> 00:19:16,549

 

598
00:19:16,549 --> 00:19:19,110

 

599
00:19:19,110 --> 00:19:20,710

 

600
00:19:20,710 --> 00:19:23,510

 

601
00:19:23,510 --> 00:19:25,029

 

602
00:19:25,029 --> 00:19:26,310

 

603
00:19:26,310 --> 00:19:27,590

 

604
00:19:27,590 --> 00:19:29,990

 

605
00:19:29,990 --> 00:19:32,789

 

606
00:19:32,789 --> 00:19:32,799

 

607
00:19:32,799 --> 00:19:37,350


608
00:19:37,350 --> 00:19:39,510

 

609
00:19:39,510 --> 00:19:41,669

 

610
00:19:41,669 --> 00:19:44,150

 

611
00:19:44,150 --> 00:19:45,909

 

612
00:19:45,909 --> 00:19:47,029

 

613
00:19:47,029 --> 00:19:50,470

 

614
00:19:50,470 --> 00:19:52,789

 

615
00:19:52,789 --> 00:19:52,799

 

616
00:19:52,799 --> 00:19:53,750


617
00:19:53,750 --> 00:19:57,909

 

618
00:19:57,909 --> 00:20:00,630

 

619
00:20:00,630 --> 00:20:02,870

 

620
00:20:02,870 --> 00:20:04,390

 

621
00:20:04,390 --> 00:20:06,950

 

622
00:20:06,950 --> 00:20:08,950

 

623
00:20:08,950 --> 00:20:10,710

 

624
00:20:10,710 --> 00:20:12,630

 

625
00:20:12,630 --> 00:20:14,789

 

626
00:20:14,789 --> 00:20:14,799

 

627
00:20:14,799 --> 00:20:15,830


628
00:20:15,830 --> 00:20:17,909

 

629
00:20:17,909 --> 00:20:19,350

 

630
00:20:19,350 --> 00:20:19,360

 

631
00:20:19,360 --> 00:20:20,470


632
00:20:20,470 --> 00:20:22,149

 

633
00:20:22,149 --> 00:20:23,270

 

634
00:20:23,270 --> 00:20:24,630

 

635
00:20:24,630 --> 00:20:28,149

 

636
00:20:28,149 --> 00:20:30,789

 

637
00:20:30,789 --> 00:20:33,029

 

638
00:20:33,029 --> 00:20:34,310

 

639
00:20:34,310 --> 00:20:36,549

 

640
00:20:36,549 --> 00:20:39,110

 

641
00:20:39,110 --> 00:20:40,950

 

642
00:20:40,950 --> 00:20:42,470

 

643
00:20:42,470 --> 00:20:45,669

 

644
00:20:45,669 --> 00:20:48,950

 

645
00:20:48,950 --> 00:20:51,750

 

646
00:20:51,750 --> 00:20:53,430

 

647
00:20:53,430 --> 00:20:55,750

 

648
00:20:55,750 --> 00:20:57,669

 

649
00:20:57,669 --> 00:20:57,679

 

650
00:20:57,679 --> 00:20:58,470


651
00:20:58,470 --> 00:21:00,149

 

652
00:21:00,149 --> 00:21:01,590

 

653
00:21:01,590 --> 00:21:03,669

 

654
00:21:03,669 --> 00:21:03,679

 

655
00:21:03,679 --> 00:21:06,549


656
00:21:06,549 --> 00:21:07,750

 

657
00:21:07,750 --> 00:21:10,149

 

658
00:21:10,149 --> 00:21:12,310

 

659
00:21:12,310 --> 00:21:12,320

 

660
00:21:12,320 --> 00:21:13,270


661
00:21:13,270 --> 00:21:16,470

 

662
00:21:16,470 --> 00:21:18,950

 

663
00:21:18,950 --> 00:21:21,430

 

664
00:21:21,430 --> 00:21:23,350

 

665
00:21:23,350 --> 00:21:24,549

 

666
00:21:24,549 --> 00:21:26,549

 

667
00:21:26,549 --> 00:21:28,310

 

668
00:21:28,310 --> 00:21:30,549

 

669
00:21:30,549 --> 00:21:31,990

 

670
00:21:31,990 --> 00:21:32,000

 

671
00:21:32,000 --> 00:21:32,950


672
00:21:32,950 --> 00:21:34,870

 

673
00:21:34,870 --> 00:21:36,630

 

674
00:21:36,630 --> 00:21:39,110

 

675
00:21:39,110 --> 00:21:41,270

 

676
00:21:41,270 --> 00:21:43,029

 

677
00:21:43,029 --> 00:21:45,909

 

678
00:21:45,909 --> 00:21:48,870

 

679
00:21:48,870 --> 00:21:53,029

 

680
00:21:53,029 --> 00:21:53,039

 

681
00:21:53,039 --> 00:21:55,990


682
00:21:55,990 --> 00:22:05,270

 

683
00:22:05,270 --> 00:22:07,190

 

684
00:22:07,190 --> 00:22:10,070

 

685
00:22:10,070 --> 00:22:10,080

 

686
00:22:10,080 --> 00:22:12,149


687
00:22:12,149 --> 00:22:15,590

 

688
00:22:15,590 --> 00:22:16,950

 

689
00:22:16,950 --> 00:22:19,430

 

690
00:22:19,430 --> 00:22:22,470

 

691
00:22:22,470 --> 00:22:24,230

 

692
00:22:24,230 --> 00:22:26,870

 

693
00:22:26,870 --> 00:22:29,350

 

694
00:22:29,350 --> 00:22:32,630

 

695
00:22:32,630 --> 00:22:32,640

 

696
00:22:32,640 --> 00:22:33,510


697
00:22:33,510 --> 00:22:35,110

 

698
00:22:35,110 --> 00:22:37,430

 

699
00:22:37,430 --> 00:22:39,270

 

700
00:22:39,270 --> 00:22:41,270

 

701
00:22:41,270 --> 00:22:44,149

 

702
00:22:44,149 --> 00:22:46,070

 

703
00:22:46,070 --> 00:22:49,669

 

704
00:22:49,669 --> 00:22:51,430

 

705
00:22:51,430 --> 00:22:53,590

 

706
00:22:53,590 --> 00:22:57,110

 

707
00:22:57,110 --> 00:22:57,120

 

708
00:22:57,120 --> 00:23:00,470


709
00:23:00,470 --> 00:23:00,480

 

710
00:23:00,480 --> 00:23:03,350


711
00:23:03,350 --> 00:23:04,710

 

712
00:23:04,710 --> 00:23:06,390

 

713
00:23:06,390 --> 00:23:09,029

 

714
00:23:09,029 --> 00:23:11,270

 

715
00:23:11,270 --> 00:23:13,590

 

716
00:23:13,590 --> 00:23:17,110

 

717
00:23:17,110 --> 00:23:18,789

 

718
00:23:18,789 --> 00:23:20,310

 

719
00:23:20,310 --> 00:23:23,350

 

720
00:23:23,350 --> 00:23:25,029

 

721
00:23:25,029 --> 00:23:27,510

 

722
00:23:27,510 --> 00:23:29,590

 

723
00:23:29,590 --> 00:23:31,350

 

724
00:23:31,350 --> 00:23:33,590

 

725
00:23:33,590 --> 00:23:35,029

 

726
00:23:35,029 --> 00:23:40,310

 

727
00:23:40,310 --> 00:23:42,070

 

728
00:23:42,070 --> 00:23:43,350

 

729
00:23:43,350 --> 00:23:44,870

 

730
00:23:44,870 --> 00:23:46,710

 

731
00:23:46,710 --> 00:23:47,830

 

732
00:23:47,830 --> 00:23:49,029

 

733
00:23:49,029 --> 00:23:51,029

 

734
00:23:51,029 --> 00:23:52,470

 

735
00:23:52,470 --> 00:23:54,950

 

736
00:23:54,950 --> 00:23:56,630

 

737
00:23:56,630 --> 00:23:56,640

 

738
00:23:56,640 --> 00:23:58,789


739
00:23:58,789 --> 00:24:00,870

 

740
00:24:00,870 --> 00:24:02,870

 

741
00:24:02,870 --> 00:24:05,750

 

742
00:24:05,750 --> 00:24:08,310

 

743
00:24:08,310 --> 00:24:11,430

 

744
00:24:11,430 --> 00:24:13,190

 

745
00:24:13,190 --> 00:24:14,549

 

746
00:24:14,549 --> 00:24:15,830

 

747
00:24:15,830 --> 00:24:19,430

 

748
00:24:19,430 --> 00:24:19,440

 

749
00:24:19,440 --> 00:24:20,230


750
00:24:20,230 --> 00:24:22,230

 

751
00:24:22,230 --> 00:24:24,070

 

752
00:24:24,070 --> 00:24:26,230

 

753
00:24:26,230 --> 00:24:29,590

 

754
00:24:29,590 --> 00:24:29,600

 

755
00:24:29,600 --> 00:24:30,310


756
00:24:30,310 --> 00:24:33,750

 

757
00:24:33,750 --> 00:24:37,909

 

758
00:24:37,909 --> 00:24:40,470

 

759
00:24:40,470 --> 00:24:44,830

 

760
00:24:44,830 --> 00:24:44,840

 

761
00:24:44,840 --> 00:24:46,630


762
00:24:46,630 --> 00:24:48,710

 

763
00:24:48,710 --> 00:24:50,630

 

764
00:24:50,630 --> 00:24:50,640

 

765
00:24:50,640 --> 00:24:51,000


766
00:24:51,000 --> 00:24:51,010

 

767
00:24:51,010 --> 00:24:52,470


768
00:24:52,470 --> 00:24:54,149

 

769
00:24:54,149 --> 00:24:56,230

 

770
00:24:56,230 --> 00:24:58,630

 

771
00:24:58,630 --> 00:25:00,950

 

772
00:25:00,950 --> 00:25:03,990

 

773
00:25:03,990 --> 00:25:05,669

 

774
00:25:05,669 --> 00:25:05,679

 

775
00:25:05,679 --> 00:25:08,070


776
00:25:08,070 --> 00:25:09,750

 

777
00:25:09,750 --> 00:25:12,310

 

778
00:25:12,310 --> 00:25:13,990

 

779
00:25:13,990 --> 00:25:16,470

 

780
00:25:16,470 --> 00:25:18,789

 

781
00:25:18,789 --> 00:25:21,190

 

782
00:25:21,190 --> 00:25:23,510

 

783
00:25:23,510 --> 00:25:25,909

 

784
00:25:25,909 --> 00:25:25,919

 

785
00:25:25,919 --> 00:25:26,950


786
00:25:26,950 --> 00:25:29,269

 

787
00:25:29,269 --> 00:25:31,430

 

788
00:25:31,430 --> 00:25:33,830

 

789
00:25:33,830 --> 00:25:35,750

 

790
00:25:35,750 --> 00:25:37,669

 

791
00:25:37,669 --> 00:25:39,510

 

792
00:25:39,510 --> 00:25:39,520

 

793
00:25:39,520 --> 00:25:40,549


794
00:25:40,549 --> 00:25:43,909

 

795
00:25:43,909 --> 00:25:43,919

 

796
00:25:43,919 --> 00:25:47,830


797
00:25:47,830 --> 00:25:49,430

 

798
00:25:49,430 --> 00:25:51,590

 

799
00:25:51,590 --> 00:25:53,430

 

800
00:25:53,430 --> 00:25:56,710

 

801
00:25:56,710 --> 00:25:58,230

 

802
00:25:58,230 --> 00:26:00,390

 

803
00:26:00,390 --> 00:26:02,470

 

804
00:26:02,470 --> 00:26:04,870

 

805
00:26:04,870 --> 00:26:07,190

 

806
00:26:07,190 --> 00:26:10,230

 

807
00:26:10,230 --> 00:26:11,350

 

808
00:26:11,350 --> 00:26:11,360

 

809
00:26:11,360 --> 00:26:14,390


810
00:26:14,390 --> 00:26:16,070

 

811
00:26:16,070 --> 00:26:17,029

 

812
00:26:17,029 --> 00:26:18,710

 

813
00:26:18,710 --> 00:26:21,830

 

814
00:26:21,830 --> 00:26:25,350

 

815
00:26:25,350 --> 00:26:25,360

 

816
00:26:25,360 --> 00:26:26,789


817
00:26:26,789 --> 00:26:28,630

 

818
00:26:28,630 --> 00:26:30,070

 

819
00:26:30,070 --> 00:26:32,310

 

820
00:26:32,310 --> 00:26:34,630

 

821
00:26:34,630 --> 00:26:36,149

 

822
00:26:36,149 --> 00:26:38,390

 

823
00:26:38,390 --> 00:26:40,070

 

824
00:26:40,070 --> 00:26:42,950

 

825
00:26:42,950 --> 00:26:45,830

 

826
00:26:45,830 --> 00:26:45,840

 

827
00:26:45,840 --> 00:26:49,190


828
00:26:49,190 --> 00:26:50,470

 

829
00:26:50,470 --> 00:26:52,789

 

830
00:26:52,789 --> 00:26:56,710

 

831
00:26:56,710 --> 00:26:58,149

 

832
00:26:58,149 --> 00:27:02,710

 

833
00:27:02,710 --> 00:27:02,720

 

834
00:27:02,720 --> 00:27:04,470


835
00:27:04,470 --> 00:27:06,630

 

836
00:27:06,630 --> 00:27:06,640

 

837
00:27:06,640 --> 00:27:08,870


838
00:27:08,870 --> 00:27:10,470

 

839
00:27:10,470 --> 00:27:11,909

 

840
00:27:11,909 --> 00:27:14,230

 

841
00:27:14,230 --> 00:27:15,990

 

842
00:27:15,990 --> 00:27:18,149

 

843
00:27:18,149 --> 00:27:20,549

 

844
00:27:20,549 --> 00:27:21,430

 

845
00:27:21,430 --> 00:27:23,990

 

846
00:27:23,990 --> 00:27:26,149

 

847
00:27:26,149 --> 00:27:28,870

 

848
00:27:28,870 --> 00:27:30,950

 

849
00:27:30,950 --> 00:27:32,870

 

850
00:27:32,870 --> 00:27:35,269

 

851
00:27:35,269 --> 00:27:38,149

 

852
00:27:38,149 --> 00:27:40,149

 

853
00:27:40,149 --> 00:27:42,230

 

854
00:27:42,230 --> 00:27:42,240

 

855
00:27:42,240 --> 00:27:46,549


856
00:27:46,549 --> 00:27:48,310

 

857
00:27:48,310 --> 00:27:49,909

 

858
00:27:49,909 --> 00:27:51,830

 

859
00:27:51,830 --> 00:27:54,710

 

860
00:27:54,710 --> 00:27:56,950

 

861
00:27:56,950 --> 00:27:59,430

 

862
00:27:59,430 --> 00:28:01,669

 

863
00:28:01,669 --> 00:28:04,070

 

864
00:28:04,070 --> 00:28:05,110

 

865
00:28:05,110 --> 00:28:09,990

 

866
00:28:09,990 --> 00:28:11,909

 

867
00:28:11,909 --> 00:28:13,590

 

868
00:28:13,590 --> 00:28:17,190

 

869
00:28:17,190 --> 00:28:19,909

 

870
00:28:19,909 --> 00:28:21,430

 

871
00:28:21,430 --> 00:28:24,310

 

872
00:28:24,310 --> 00:28:27,590

 

873
00:28:27,590 --> 00:28:28,950

 

874
00:28:28,950 --> 00:28:30,310

 

875
00:28:30,310 --> 00:28:31,510

 

876
00:28:31,510 --> 00:28:33,029

 

877
00:28:33,029 --> 00:28:36,149

 

878
00:28:36,149 --> 00:28:37,590

 

879
00:28:37,590 --> 00:28:39,590

 

880
00:28:39,590 --> 00:28:44,070

 

881
00:28:44,070 --> 00:28:45,590

 

882
00:28:45,590 --> 00:28:48,310

 

883
00:28:48,310 --> 00:28:50,310

 

884
00:28:50,310 --> 00:28:52,549

 

885
00:28:52,549 --> 00:28:54,230

 

886
00:28:54,230 --> 00:28:56,710

 

887
00:28:56,710 --> 00:29:04,870

 

888
00:29:04,870 --> 00:29:07,430

 

889
00:29:07,430 --> 00:29:08,950

 

890
00:29:08,950 --> 00:29:10,630

 

891
00:29:10,630 --> 00:29:10,640

 

892
00:29:10,640 --> 00:29:11,590


893
00:29:11,590 --> 00:29:13,669

 

894
00:29:13,669 --> 00:29:13,679

 

895
00:29:13,679 --> 00:29:15,990


896
00:29:15,990 --> 00:29:16,000

 

897
00:29:16,000 --> 00:29:18,389


898
00:29:18,389 --> 00:29:18,399

 

899
00:29:18,399 --> 00:29:19,430


900
00:29:19,430 --> 00:29:20,950

 

901
00:29:20,950 --> 00:29:22,630

 

902
00:29:22,630 --> 00:29:24,549

 

903
00:29:24,549 --> 00:29:26,470

 

904
00:29:26,470 --> 00:29:29,269

 

905
00:29:29,269 --> 00:29:30,710

 

906
00:29:30,710 --> 00:29:30,720

 

907
00:29:30,720 --> 00:29:31,510


908
00:29:31,510 --> 00:29:31,520

 

909
00:29:31,520 --> 00:29:32,950


910
00:29:32,950 --> 00:29:35,510

 

911
00:29:35,510 --> 00:29:37,590

 

912
00:29:37,590 --> 00:29:39,269

 

913
00:29:39,269 --> 00:29:40,630

 

914
00:29:40,630 --> 00:29:41,669

 

915
00:29:41,669 --> 00:29:44,070

 

916
00:29:44,070 --> 00:29:47,190

 

917
00:29:47,190 --> 00:29:47,200

 

918
00:29:47,200 --> 00:29:48,310


919
00:29:48,310 --> 00:29:49,669

 

920
00:29:49,669 --> 00:29:52,549

 

921
00:29:52,549 --> 00:29:54,950

 

922
00:29:54,950 --> 00:29:57,029

 

923
00:29:57,029 --> 00:29:59,190

 

924
00:29:59,190 --> 00:30:02,310

 

925
00:30:02,310 --> 00:30:05,190

 

926
00:30:05,190 --> 00:30:10,149

 

927
00:30:10,149 --> 00:30:12,549

 

928
00:30:12,549 --> 00:30:15,510

 

929
00:30:15,510 --> 00:30:15,520

 

930
00:30:15,520 --> 00:30:16,549


931
00:30:16,549 --> 00:30:19,350

 

932
00:30:19,350 --> 00:30:21,669

 

933
00:30:21,669 --> 00:30:25,830

 

934
00:30:25,830 --> 00:30:28,149

 

935
00:30:28,149 --> 00:30:30,549

 

936
00:30:30,549 --> 00:30:32,950

 

937
00:30:32,950 --> 00:30:34,950

 

938
00:30:34,950 --> 00:30:38,950

 

939
00:30:38,950 --> 00:30:41,269

 

940
00:30:41,269 --> 00:30:44,470

 

941
00:30:44,470 --> 00:30:47,190

 

942
00:30:47,190 --> 00:30:50,310

 

943
00:30:50,310 --> 00:30:52,950

 

944
00:30:52,950 --> 00:30:55,029

 

945
00:30:55,029 --> 00:30:57,269

 

946
00:30:57,269 --> 00:30:59,590

 

947
00:30:59,590 --> 00:31:02,310

 

948
00:31:02,310 --> 00:31:03,750

 

949
00:31:03,750 --> 00:31:06,870

 

950
00:31:06,870 --> 00:31:09,190

 

951
00:31:09,190 --> 00:31:10,630

 

952
00:31:10,630 --> 00:31:12,630

 

953
00:31:12,630 --> 00:31:14,870

 

954
00:31:14,870 --> 00:31:17,509

 

955
00:31:17,509 --> 00:31:19,110

 

956
00:31:19,110 --> 00:31:20,230

 

957
00:31:20,230 --> 00:31:20,240

 

958
00:31:20,240 --> 00:31:21,190


959
00:31:21,190 --> 00:31:23,269

 

960
00:31:23,269 --> 00:31:25,590

 

961
00:31:25,590 --> 00:31:28,230

 

962
00:31:28,230 --> 00:31:30,950

 

963
00:31:30,950 --> 00:31:35,590

 

964
00:31:35,590 --> 00:31:37,909

 

965
00:31:37,909 --> 00:31:40,149

 

966
00:31:40,149 --> 00:31:41,990

 

967
00:31:41,990 --> 00:31:43,909

 

968
00:31:43,909 --> 00:31:45,590

 

969
00:31:45,590 --> 00:31:46,789

 

970
00:31:46,789 --> 00:31:48,230

 

971
00:31:48,230 --> 00:31:48,240

 

972
00:31:48,240 --> 00:31:48,950


973
00:31:48,950 --> 00:31:50,710

 

974
00:31:50,710 --> 00:31:52,950

 

975
00:31:52,950 --> 00:31:54,710

 

976
00:31:54,710 --> 00:31:56,389

 

977
00:31:56,389 --> 00:31:57,990

 

978
00:31:57,990 --> 00:32:00,149

 

979
00:32:00,149 --> 00:32:01,590

 

980
00:32:01,590 --> 00:32:01,600

 

981
00:32:01,600 --> 00:32:04,149


982
00:32:04,149 --> 00:32:06,950

 

983
00:32:06,950 --> 00:32:08,470

 

984
00:32:08,470 --> 00:32:10,310

 

985
00:32:10,310 --> 00:32:12,710

 

986
00:32:12,710 --> 00:32:15,350

 

987
00:32:15,350 --> 00:32:17,430

 

988
00:32:17,430 --> 00:32:19,029

 

989
00:32:19,029 --> 00:32:19,990

 

990
00:32:19,990 --> 00:32:21,750

 

991
00:32:21,750 --> 00:32:23,909

 

992
00:32:23,909 --> 00:32:26,310

 

993
00:32:26,310 --> 00:32:26,320

 

994
00:32:26,320 --> 00:32:27,269


995
00:32:27,269 --> 00:32:31,110

 

996
00:32:31,110 --> 00:32:31,120

 

997
00:32:31,120 --> 00:32:33,110


998
00:32:33,110 --> 00:32:35,750

 

999
00:32:35,750 --> 00:32:38,149

 

1000
00:32:38,149 --> 00:32:40,070

 

1001
00:32:40,070 --> 00:32:41,669

 

1002
00:32:41,669 --> 00:32:41,679

 

1003
00:32:41,679 --> 00:32:43,909


1004
00:32:43,909 --> 00:32:47,430

 

1005
00:32:47,430 --> 00:32:49,029

 

1006
00:32:49,029 --> 00:32:50,230

 

1007
00:32:50,230 --> 00:32:51,430

 

1008
00:32:51,430 --> 00:32:53,350

 

1009
00:32:53,350 --> 00:32:56,950

 

1010
00:32:56,950 --> 00:32:59,669

 

1011
00:32:59,669 --> 00:33:02,470

 

1012
00:33:02,470 --> 00:33:04,549

 

1013
00:33:04,549 --> 00:33:06,230

 

1014
00:33:06,230 --> 00:33:10,549

 

1015
00:33:10,549 --> 00:33:10,559

 

1016
00:33:10,559 --> 00:33:13,269


1017
00:33:13,269 --> 00:33:15,750

 

1018
00:33:15,750 --> 00:33:15,760

 

1019
00:33:15,760 --> 00:33:17,909


1020
00:33:17,909 --> 00:33:17,919

 

1021
00:33:17,919 --> 00:33:18,710


1022
00:33:18,710 --> 00:33:21,029

 

1023
00:33:21,029 --> 00:33:25,990

 

1024
00:33:25,990 --> 00:33:27,269

 

1025
00:33:27,269 --> 00:33:28,389

 

1026
00:33:28,389 --> 00:33:29,990

 

1027
00:33:29,990 --> 00:33:30,000

 

1028
00:33:30,000 --> 00:33:30,630


1029
00:33:30,630 --> 00:33:32,630

 

1030
00:33:32,630 --> 00:33:34,950

 

1031
00:33:34,950 --> 00:33:36,389

 

1032
00:33:36,389 --> 00:33:39,110

 

1033
00:33:39,110 --> 00:33:41,110

 

1034
00:33:41,110 --> 00:33:42,789

 

1035
00:33:42,789 --> 00:33:42,799

 

1036
00:33:42,799 --> 00:33:43,669


1037
00:33:43,669 --> 00:33:43,679

 

1038
00:33:43,679 --> 00:33:45,350


1039
00:33:45,350 --> 00:33:47,669

 

1040
00:33:47,669 --> 00:33:49,669

 

1041
00:33:49,669 --> 00:33:52,789

 

1042
00:33:52,789 --> 00:33:55,190

 

1043
00:33:55,190 --> 00:33:57,190

 

1044
00:33:57,190 --> 00:33:59,269

 

1045
00:33:59,269 --> 00:33:59,279

 

1046
00:33:59,279 --> 00:34:00,789


1047
00:34:00,789 --> 00:34:02,710

 

1048
00:34:02,710 --> 00:34:05,269

 

1049
00:34:05,269 --> 00:34:08,629

 

1050
00:34:08,629 --> 00:34:09,909

 

1051
00:34:09,909 --> 00:34:11,589

 

1052
00:34:11,589 --> 00:34:13,190

 

1053
00:34:13,190 --> 00:34:13,200

 

1054
00:34:13,200 --> 00:34:13,990


1055
00:34:13,990 --> 00:34:15,109

 

1056
00:34:15,109 --> 00:34:15,119

 

1057
00:34:15,119 --> 00:34:16,950


1058
00:34:16,950 --> 00:34:18,869

 

1059
00:34:18,869 --> 00:34:21,349

 

1060
00:34:21,349 --> 00:34:25,430

 

1061
00:34:25,430 --> 00:34:27,589

 

1062
00:34:27,589 --> 00:34:29,750

 

1063
00:34:29,750 --> 00:34:30,950

 

1064
00:34:30,950 --> 00:34:32,629

 

1065
00:34:32,629 --> 00:34:34,389

 

1066
00:34:34,389 --> 00:34:36,230

 

1067
00:34:36,230 --> 00:34:38,149

 

1068
00:34:38,149 --> 00:34:40,069

 

1069
00:34:40,069 --> 00:34:42,230

 

1070
00:34:42,230 --> 00:34:43,190

 

1071
00:34:43,190 --> 00:34:43,200

 

1072
00:34:43,200 --> 00:34:45,750


1073
00:34:45,750 --> 00:34:47,190

 

1074
00:34:47,190 --> 00:34:49,109

 

1075
00:34:49,109 --> 00:34:51,030

 

1076
00:34:51,030 --> 00:34:53,349

 

1077
00:34:53,349 --> 00:34:55,190

 

1078
00:34:55,190 --> 00:34:56,389

 

1079
00:34:56,389 --> 00:34:58,069

 

1080
00:34:58,069 --> 00:34:59,829

 

1081
00:34:59,829 --> 00:34:59,839

 

1082
00:34:59,839 --> 00:35:00,550


1083
00:35:00,550 --> 00:35:02,950

 

1084
00:35:02,950 --> 00:35:06,310

 

1085
00:35:06,310 --> 00:35:09,190

 

1086
00:35:09,190 --> 00:35:11,430

 

1087
00:35:11,430 --> 00:35:13,190

 

1088
00:35:13,190 --> 00:35:17,670

 

1089
00:35:17,670 --> 00:35:19,670

 

1090
00:35:19,670 --> 00:35:22,230

 

1091
00:35:22,230 --> 00:35:24,550

 

1092
00:35:24,550 --> 00:35:27,589

 

1093
00:35:27,589 --> 00:35:28,790

 

1094
00:35:28,790 --> 00:35:31,030

 

1095
00:35:31,030 --> 00:35:33,030

 

1096
00:35:33,030 --> 00:35:35,430

 

1097
00:35:35,430 --> 00:35:37,829

 

1098
00:35:37,829 --> 00:35:39,990

 

1099
00:35:39,990 --> 00:35:41,430

 

1100
00:35:41,430 --> 00:35:43,589

 

1101
00:35:43,589 --> 00:35:46,150

 

1102
00:35:46,150 --> 00:35:48,710

 

1103
00:35:48,710 --> 00:35:51,030

 

1104
00:35:51,030 --> 00:35:51,040

 

1105
00:35:51,040 --> 00:35:52,950


1106
00:35:52,950 --> 00:35:54,950

 

1107
00:35:54,950 --> 00:35:56,390

 

1108
00:35:56,390 --> 00:35:58,710

 

1109
00:35:58,710 --> 00:36:00,390

 

1110
00:36:00,390 --> 00:36:03,030

 

1111
00:36:03,030 --> 00:36:04,470

 

1112
00:36:04,470 --> 00:36:06,310

 

1113
00:36:06,310 --> 00:36:06,320

 

1114
00:36:06,320 --> 00:36:07,190


1115
00:36:07,190 --> 00:36:09,589

 

1116
00:36:09,589 --> 00:36:11,990

 

1117
00:36:11,990 --> 00:36:13,990

 

1118
00:36:13,990 --> 00:36:16,069

 

1119
00:36:16,069 --> 00:36:17,430

 

1120
00:36:17,430 --> 00:36:19,349

 

1121
00:36:19,349 --> 00:36:20,310

 

1122
00:36:20,310 --> 00:36:22,470

 

1123
00:36:22,470 --> 00:36:23,589

 

1124
00:36:23,589 --> 00:36:25,430

 

1125
00:36:25,430 --> 00:36:25,440

 

1126
00:36:25,440 --> 00:36:26,630


1127
00:36:26,630 --> 00:36:26,640

 

1128
00:36:26,640 --> 00:36:27,910


1129
00:36:27,910 --> 00:36:27,920

 

1130
00:36:27,920 --> 00:36:30,390


1131
00:36:30,390 --> 00:36:30,400

 

1132
00:36:30,400 --> 00:36:32,150


1133
00:36:32,150 --> 00:36:33,270

 

1134
00:36:33,270 --> 00:36:37,270

 

1135
00:36:37,270 --> 00:36:39,510

 

1136
00:36:39,510 --> 00:36:43,990

 

1137
00:36:43,990 --> 00:36:46,870

 

1138
00:36:46,870 --> 00:36:48,310

 

1139
00:36:48,310 --> 00:36:50,950

 

1140
00:36:50,950 --> 00:36:53,349

 

1141
00:36:53,349 --> 00:36:54,390

 

1142
00:36:54,390 --> 00:36:55,910

 

1143
00:36:55,910 --> 00:36:57,990

 

1144
00:36:57,990 --> 00:36:58,000

 

1145
00:36:58,000 --> 00:36:59,349


1146
00:36:59,349 --> 00:37:02,550

 

1147
00:37:02,550 --> 00:37:04,150

 

1148
00:37:04,150 --> 00:37:05,589

 

1149
00:37:05,589 --> 00:37:07,750

 

1150
00:37:07,750 --> 00:37:07,760

 

1151
00:37:07,760 --> 00:37:08,870


1152
00:37:08,870 --> 00:37:11,510

 

1153
00:37:11,510 --> 00:37:11,520

 

1154
00:37:11,520 --> 00:37:13,270


1155
00:37:13,270 --> 00:37:14,550

 

1156
00:37:14,550 --> 00:37:16,310

 

1157
00:37:16,310 --> 00:37:17,829

 

1158
00:37:17,829 --> 00:37:19,829

 

1159
00:37:19,829 --> 00:37:22,310

 

1160
00:37:22,310 --> 00:37:23,910

 

1161
00:37:23,910 --> 00:37:26,390

 

1162
00:37:26,390 --> 00:37:28,230

 

1163
00:37:28,230 --> 00:37:28,240

 

1164
00:37:28,240 --> 00:37:29,750


1165
00:37:29,750 --> 00:37:31,750

 

1166
00:37:31,750 --> 00:37:33,990

 

1167
00:37:33,990 --> 00:37:34,000

 

1168
00:37:34,000 --> 00:37:35,109


1169
00:37:35,109 --> 00:37:39,109

 

1170
00:37:39,109 --> 00:37:40,710

 

1171
00:37:40,710 --> 00:37:42,550

 

1172
00:37:42,550 --> 00:37:42,560

 

1173
00:37:42,560 --> 00:37:43,510


1174
00:37:43,510 --> 00:37:43,520

 

1175
00:37:43,520 --> 00:37:46,470


1176
00:37:46,470 --> 00:37:48,069

 

1177
00:37:48,069 --> 00:37:49,270

 

1178
00:37:49,270 --> 00:37:51,430

 

1179
00:37:51,430 --> 00:37:53,750

 

1180
00:37:53,750 --> 00:37:55,109

 

1181
00:37:55,109 --> 00:37:56,390

 

1182
00:37:56,390 --> 00:37:59,430

 

1183
00:37:59,430 --> 00:38:01,430

 

1184
00:38:01,430 --> 00:38:03,030

 

1185
00:38:03,030 --> 00:38:04,790

 

1186
00:38:04,790 --> 00:38:07,750

 

1187
00:38:07,750 --> 00:38:07,760

 

1188
00:38:07,760 --> 00:38:09,030


1189
00:38:09,030 --> 00:38:11,190

 

1190
00:38:11,190 --> 00:38:12,310

 

1191
00:38:12,310 --> 00:38:12,320

 

1192
00:38:12,320 --> 00:38:14,950


1193
00:38:14,950 --> 00:38:14,960

 

1194
00:38:14,960 --> 00:38:16,470


1195
00:38:16,470 --> 00:38:18,630

 

1196
00:38:18,630 --> 00:38:21,109

 

1197
00:38:21,109 --> 00:38:23,030

 

1198
00:38:23,030 --> 00:38:25,109

 

1199
00:38:25,109 --> 00:38:27,589

 

1200
00:38:27,589 --> 00:38:29,430

 

1201
00:38:29,430 --> 00:38:30,950

 

1202
00:38:30,950 --> 00:38:32,390

 

1203
00:38:32,390 --> 00:38:36,470

 

1204
00:38:36,470 --> 00:38:40,630

 

1205
00:38:40,630 --> 00:38:42,790

 

1206
00:38:42,790 --> 00:38:45,589

 

1207
00:38:45,589 --> 00:38:47,670

 

1208
00:38:47,670 --> 00:38:49,670

 

1209
00:38:49,670 --> 00:38:51,109

 

1210
00:38:51,109 --> 00:38:53,510

 

1211
00:38:53,510 --> 00:38:55,190

 

1212
00:38:55,190 --> 00:38:56,150

 

1213
00:38:56,150 --> 00:38:59,190

 

1214
00:38:59,190 --> 00:39:01,109

 

1215
00:39:01,109 --> 00:39:04,230

 

1216
00:39:04,230 --> 00:39:06,150

 

1217
00:39:06,150 --> 00:39:09,430

 

1218
00:39:09,430 --> 00:39:11,750

 

1219
00:39:11,750 --> 00:39:14,390

 

1220
00:39:14,390 --> 00:39:16,870

 

1221
00:39:16,870 --> 00:39:20,630

 

1222
00:39:20,630 --> 00:39:22,390

 

1223
00:39:22,390 --> 00:39:24,230

 

1224
00:39:24,230 --> 00:39:28,069

 

1225
00:39:28,069 --> 00:39:30,150

 

1226
00:39:30,150 --> 00:39:32,470

 

1227
00:39:32,470 --> 00:39:32,480

 

1228
00:39:32,480 --> 00:39:33,430


1229
00:39:33,430 --> 00:39:35,990

 

1230
00:39:35,990 --> 00:39:38,630

 

1231
00:39:38,630 --> 00:39:40,630

 

1232
00:39:40,630 --> 00:39:43,109

 

1233
00:39:43,109 --> 00:39:44,870

 

1234
00:39:44,870 --> 00:39:46,950

 

1235
00:39:46,950 --> 00:39:49,589

 

1236
00:39:49,589 --> 00:39:49,599

 

1237
00:39:49,599 --> 00:39:50,550


1238
00:39:50,550 --> 00:39:52,390

 

1239
00:39:52,390 --> 00:39:53,910

 

1240
00:39:53,910 --> 00:39:55,030

 

1241
00:39:55,030 --> 00:39:56,550

 

1242
00:39:56,550 --> 00:39:59,349

 

1243
00:39:59,349 --> 00:40:01,910

 

1244
00:40:01,910 --> 00:40:02,950

 

1245
00:40:02,950 --> 00:40:02,960

 

1246
00:40:02,960 --> 00:40:04,150


1247
00:40:04,150 --> 00:40:06,390

 

1248
00:40:06,390 --> 00:40:08,470

 

1249
00:40:08,470 --> 00:40:10,950

 

1250
00:40:10,950 --> 00:40:12,790

 

1251
00:40:12,790 --> 00:40:14,630

 

1252
00:40:14,630 --> 00:40:16,230

 

1253
00:40:16,230 --> 00:40:17,670

 

1254
00:40:17,670 --> 00:40:19,829

 

1255
00:40:19,829 --> 00:40:20,870

 

1256
00:40:20,870 --> 00:40:22,390

 

1257
00:40:22,390 --> 00:40:25,589

 

1258
00:40:25,589 --> 00:40:27,270

 

1259
00:40:27,270 --> 00:40:27,280

 

1260
00:40:27,280 --> 00:40:28,069


1261
00:40:28,069 --> 00:40:30,550

 

1262
00:40:30,550 --> 00:40:32,390

 

1263
00:40:32,390 --> 00:40:34,550

 

1264
00:40:34,550 --> 00:40:36,550

 

1265
00:40:36,550 --> 00:40:38,630

 

1266
00:40:38,630 --> 00:40:40,309

 

1267
00:40:40,309 --> 00:40:42,470

 

1268
00:40:42,470 --> 00:40:44,550

 

1269
00:40:44,550 --> 00:40:45,750

 

1270
00:40:45,750 --> 00:40:47,829

 

1271
00:40:47,829 --> 00:40:51,589

 

1272
00:40:51,589 --> 00:40:53,349

 

1273
00:40:53,349 --> 00:40:54,870

 

1274
00:40:54,870 --> 00:40:57,430

 

1275
00:40:57,430 --> 00:40:59,190

 

1276
00:40:59,190 --> 00:41:00,550

 

1277
00:41:00,550 --> 00:41:02,390

 

1278
00:41:02,390 --> 00:41:04,309

 

1279
00:41:04,309 --> 00:41:07,670

 

1280
00:41:07,670 --> 00:41:07,680

 

1281
00:41:07,680 --> 00:41:08,790


1282
00:41:08,790 --> 00:41:08,800

 

1283
00:41:08,800 --> 00:41:10,150


1284
00:41:10,150 --> 00:41:10,160

 

1285
00:41:10,160 --> 00:41:11,510


1286
00:41:11,510 --> 00:41:13,910

 

1287
00:41:13,910 --> 00:41:15,750

 

1288
00:41:15,750 --> 00:41:18,030

 

1289
00:41:18,030 --> 00:41:19,750

 

1290
00:41:19,750 --> 00:41:19,760

 

1291
00:41:19,760 --> 00:41:21,910


1292
00:41:21,910 --> 00:41:23,109

 

1293
00:41:23,109 --> 00:41:25,750

 

1294
00:41:25,750 --> 00:41:28,309

 

1295
00:41:28,309 --> 00:41:28,319

 

1296
00:41:28,319 --> 00:41:31,190


1297
00:41:31,190 --> 00:41:33,109

 

1298
00:41:33,109 --> 00:41:35,190

 

1299
00:41:35,190 --> 00:41:36,630

 

1300
00:41:36,630 --> 00:41:36,640

 

1301
00:41:36,640 --> 00:41:37,430


1302
00:41:37,430 --> 00:41:40,470

 

1303
00:41:40,470 --> 00:41:42,309

 

1304
00:41:42,309 --> 00:41:45,190

 

1305
00:41:45,190 --> 00:41:48,150

 

1306
00:41:48,150 --> 00:41:49,670

 

1307
00:41:49,670 --> 00:41:51,109

 

1308
00:41:51,109 --> 00:41:53,030

 

1309
00:41:53,030 --> 00:41:55,510

 

1310
00:41:55,510 --> 00:41:56,710

 

1311
00:41:56,710 --> 00:42:00,069

 

1312
00:42:00,069 --> 00:42:01,270

 

1313
00:42:01,270 --> 00:42:05,109

 

1314
00:42:05,109 --> 00:42:07,270

 

1315
00:42:07,270 --> 00:42:09,349

 

1316
00:42:09,349 --> 00:42:11,030

 

1317
00:42:11,030 --> 00:42:12,550

 

1318
00:42:12,550 --> 00:42:14,390

 

1319
00:42:14,390 --> 00:42:16,630

 

1320
00:42:16,630 --> 00:42:18,630

 

1321
00:42:18,630 --> 00:42:20,950

 

1322
00:42:20,950 --> 00:42:23,030

 

1323
00:42:23,030 --> 00:42:23,040

 

1324
00:42:23,040 --> 00:42:24,150


1325
00:42:24,150 --> 00:42:26,950

 

1326
00:42:26,950 --> 00:42:29,270

 

1327
00:42:29,270 --> 00:42:29,280

 

1328
00:42:29,280 --> 00:42:30,630


1329
00:42:30,630 --> 00:42:33,990

 

1330
00:42:33,990 --> 00:42:34,000

 

1331
00:42:34,000 --> 00:42:35,270


1332
00:42:35,270 --> 00:42:37,190

 

1333
00:42:37,190 --> 00:42:39,910

 

1334
00:42:39,910 --> 00:42:41,750

 

1335
00:42:41,750 --> 00:42:44,150

 

1336
00:42:44,150 --> 00:42:47,270

 

1337
00:42:47,270 --> 00:42:49,270

 

1338
00:42:49,270 --> 00:42:50,870

 

1339
00:42:50,870 --> 00:42:53,349

 

1340
00:42:53,349 --> 00:42:53,359

 

1341
00:42:53,359 --> 00:42:54,470


1342
00:42:54,470 --> 00:42:57,349

 

1343
00:42:57,349 --> 00:43:00,470

 

1344
00:43:00,470 --> 00:43:01,589

 

1345
00:43:01,589 --> 00:43:03,430

 

1346
00:43:03,430 --> 00:43:05,750

 

1347
00:43:05,750 --> 00:43:08,550

 

1348
00:43:08,550 --> 00:43:10,630

 

1349
00:43:10,630 --> 00:43:12,230

 

1350
00:43:12,230 --> 00:43:13,990

 

1351
00:43:13,990 --> 00:43:15,750

 

1352
00:43:15,750 --> 00:43:18,069

 

1353
00:43:18,069 --> 00:43:20,390

 

1354
00:43:20,390 --> 00:43:23,349

 

1355
00:43:23,349 --> 00:43:25,589

 

1356
00:43:25,589 --> 00:43:25,599

 

1357
00:43:25,599 --> 00:43:26,390


1358
00:43:26,390 --> 00:43:28,950

 

1359
00:43:28,950 --> 00:43:30,470

 

1360
00:43:30,470 --> 00:43:30,480

 

1361
00:43:30,480 --> 00:43:31,430


1362
00:43:31,430 --> 00:43:31,440

 

1363
00:43:31,440 --> 00:43:33,349


1364
00:43:33,349 --> 00:43:35,349

 

1365
00:43:35,349 --> 00:43:38,069

 

1366
00:43:38,069 --> 00:43:40,309

 

1367
00:43:40,309 --> 00:43:42,790

 

1368
00:43:42,790 --> 00:43:44,150

 

1369
00:43:44,150 --> 00:43:44,160

 

1370
00:43:44,160 --> 00:43:45,349


1371
00:43:45,349 --> 00:43:45,359

 

1372
00:43:45,359 --> 00:43:46,710


1373
00:43:46,710 --> 00:43:48,150

 

1374
00:43:48,150 --> 00:43:50,309

 

1375
00:43:50,309 --> 00:43:52,069

 

1376
00:43:52,069 --> 00:43:54,230

 

1377
00:43:54,230 --> 00:43:56,550

 

1378
00:43:56,550 --> 00:43:59,030

 

1379
00:43:59,030 --> 00:44:00,630

 

1380
00:44:00,630 --> 00:44:00,640

 

1381
00:44:00,640 --> 00:44:01,750


1382
00:44:01,750 --> 00:44:03,109

 

1383
00:44:03,109 --> 00:44:05,270

 

1384
00:44:05,270 --> 00:44:08,069

 

1385
00:44:08,069 --> 00:44:10,390

 

1386
00:44:10,390 --> 00:44:12,390

 

1387
00:44:12,390 --> 00:44:14,309

 

1388
00:44:14,309 --> 00:44:16,710

 

1389
00:44:16,710 --> 00:44:18,470

 

1390
00:44:18,470 --> 00:44:21,510

 

1391
00:44:21,510 --> 00:44:23,750

 

1392
00:44:23,750 --> 00:44:23,760

 

1393
00:44:23,760 --> 00:44:24,710


1394
00:44:24,710 --> 00:44:24,720

 

1395
00:44:24,720 --> 00:44:25,750


1396
00:44:25,750 --> 00:44:29,670

 

1397
00:44:29,670 --> 00:44:31,510

 

1398
00:44:31,510 --> 00:44:33,910

 

1399
00:44:33,910 --> 00:44:35,829

 

1400
00:44:35,829 --> 00:44:37,829

 

1401
00:44:37,829 --> 00:44:39,990

 

1402
00:44:39,990 --> 00:44:42,550

 

1403
00:44:42,550 --> 00:44:46,790

 

1404
00:44:46,790 --> 00:44:46,800

 

1405
00:44:46,800 --> 00:44:48,150


1406
00:44:48,150 --> 00:44:50,470

 

1407
00:44:50,470 --> 00:44:50,480

 

1408
00:44:50,480 --> 00:44:52,870


1409
00:44:52,870 --> 00:44:52,880

 

1410
00:44:52,880 --> 00:44:55,430


1411
00:44:55,430 --> 00:44:55,440

 

1412
00:44:55,440 --> 00:44:57,270


1413
00:44:57,270 --> 00:45:00,309

 

1414
00:45:00,309 --> 00:45:03,109

 

1415
00:45:03,109 --> 00:45:05,670

 

1416
00:45:05,670 --> 00:45:07,109

 

1417
00:45:07,109 --> 00:45:08,630

 

1418
00:45:08,630 --> 00:45:09,829

 

1419
00:45:09,829 --> 00:45:11,510

 

1420
00:45:11,510 --> 00:45:14,230

 

1421
00:45:14,230 --> 00:45:15,829

 

1422
00:45:15,829 --> 00:45:15,839

 

1423
00:45:15,839 --> 00:45:16,790


1424
00:45:16,790 --> 00:45:18,790

 

1425
00:45:18,790 --> 00:45:18,800

 

1426
00:45:18,800 --> 00:45:20,230


1427
00:45:20,230 --> 00:45:22,150

 

1428
00:45:22,150 --> 00:45:22,160

 

1429
00:45:22,160 --> 00:45:24,230


1430
00:45:24,230 --> 00:45:24,240

 

1431
00:45:24,240 --> 00:45:25,270


1432
00:45:25,270 --> 00:45:25,280

 

1433
00:45:25,280 --> 00:45:26,870


1434
00:45:26,870 --> 00:45:29,750

 

1435
00:45:29,750 --> 00:45:37,030

 

1436
00:45:37,030 --> 00:45:39,990

 

1437
00:45:39,990 --> 00:45:42,230

 

1438
00:45:42,230 --> 00:45:43,430

 

1439
00:45:43,430 --> 00:45:45,510

 

1440
00:45:45,510 --> 00:45:47,190

 

1441
00:45:47,190 --> 00:45:49,109

 

1442
00:45:49,109 --> 00:45:51,030

 

1443
00:45:51,030 --> 00:45:52,230

 

1444
00:45:52,230 --> 00:45:53,589

 

1445
00:45:53,589 --> 00:45:53,599

 

1446
00:45:53,599 --> 00:45:54,870


1447
00:45:54,870 --> 00:45:57,030

 

1448
00:45:57,030 --> 00:45:57,040

 

1449
00:45:57,040 --> 00:45:58,069


1450
00:45:58,069 --> 00:45:58,079

 

1451
00:45:58,079 --> 00:45:59,109


1452
00:45:59,109 --> 00:46:02,550

 

1453
00:46:02,550 --> 00:46:04,550

 

1454
00:46:04,550 --> 00:46:06,150

 

1455
00:46:06,150 --> 00:46:08,550

 

1456
00:46:08,550 --> 00:46:11,510

 

1457
00:46:11,510 --> 00:46:14,550

 

1458
00:46:14,550 --> 00:46:18,950

 

1459
00:46:18,950 --> 00:46:21,589

 

1460
00:46:21,589 --> 00:46:23,430

 

1461
00:46:23,430 --> 00:46:25,829

 

1462
00:46:25,829 --> 00:46:27,349

 

1463
00:46:27,349 --> 00:46:28,790

 

1464
00:46:28,790 --> 00:46:32,150

 

1465
00:46:32,150 --> 00:46:33,829

 

1466
00:46:33,829 --> 00:46:36,390

 

1467
00:46:36,390 --> 00:46:38,630

 

1468
00:46:38,630 --> 00:46:40,630

 

1469
00:46:40,630 --> 00:46:42,390

 

1470
00:46:42,390 --> 00:46:43,750

 

1471
00:46:43,750 --> 00:46:48,069

 

1472
00:46:48,069 --> 00:46:48,079

 

1473
00:46:48,079 --> 00:46:49,910


1474
00:46:49,910 --> 00:46:51,349

 

1475
00:46:51,349 --> 00:46:53,270

 

1476
00:46:53,270 --> 00:46:56,550

 

1477
00:46:56,550 --> 00:46:58,470

 

1478
00:46:58,470 --> 00:46:58,480

 

1479
00:46:58,480 --> 00:46:59,270


1480
00:46:59,270 --> 00:47:01,109

 

1481
00:47:01,109 --> 00:47:03,589

 

1482
00:47:03,589 --> 00:47:06,390

 

1483
00:47:06,390 --> 00:47:07,910

 

1484
00:47:07,910 --> 00:47:09,910

 

1485
00:47:09,910 --> 00:47:09,920

 

1486
00:47:09,920 --> 00:47:10,710


1487
00:47:10,710 --> 00:47:12,870

 

1488
00:47:12,870 --> 00:47:15,670

 

1489
00:47:15,670 --> 00:47:18,870

 

1490
00:47:18,870 --> 00:47:20,470

 

1491
00:47:20,470 --> 00:47:23,589

 

1492
00:47:23,589 --> 00:47:25,910

 

1493
00:47:25,910 --> 00:47:28,549

 

1494
00:47:28,549 --> 00:47:30,710

 

1495
00:47:30,710 --> 00:47:32,549

 

1496
00:47:32,549 --> 00:47:34,870

 

1497
00:47:34,870 --> 00:47:36,069

 

1498
00:47:36,069 --> 00:47:37,910

 

1499
00:47:37,910 --> 00:47:40,150

 

1500
00:47:40,150 --> 00:47:41,750

 

1501
00:47:41,750 --> 00:47:44,150

 

1502
00:47:44,150 --> 00:47:46,630

 

1503
00:47:46,630 --> 00:47:48,230

 

1504
00:47:48,230 --> 00:47:49,910

 

1505
00:47:49,910 --> 00:47:52,309

 

1506
00:47:52,309 --> 00:47:54,230

 

1507
00:47:54,230 --> 00:47:56,069

 

1508
00:47:56,069 --> 00:47:59,589

 

1509
00:47:59,589 --> 00:48:01,109

 

1510
00:48:01,109 --> 00:48:02,950

 

1511
00:48:02,950 --> 00:48:04,870

 

1512
00:48:04,870 --> 00:48:06,630

 

1513
00:48:06,630 --> 00:48:06,640

 

1514
00:48:06,640 --> 00:48:07,910


1515
00:48:07,910 --> 00:48:09,190

 

1516
00:48:09,190 --> 00:48:11,430

 

1517
00:48:11,430 --> 00:48:12,870

 

1518
00:48:12,870 --> 00:48:14,790

 

1519
00:48:14,790 --> 00:48:18,309

 

1520
00:48:18,309 --> 00:48:20,870

 

1521
00:48:20,870 --> 00:48:24,549

 

1522
00:48:24,549 --> 00:48:26,069

 

1523
00:48:26,069 --> 00:48:28,790

 

1524
00:48:28,790 --> 00:48:29,910

 

1525
00:48:29,910 --> 00:48:32,069

 

1526
00:48:32,069 --> 00:48:32,079

 

1527
00:48:32,079 --> 00:48:33,670


1528
00:48:33,670 --> 00:48:35,750

 

1529
00:48:35,750 --> 00:48:38,069

 

1530
00:48:38,069 --> 00:48:40,230

 

1531
00:48:40,230 --> 00:48:40,240

 

1532
00:48:40,240 --> 00:48:42,069


1533
00:48:42,069 --> 00:48:44,069

 

1534
00:48:44,069 --> 00:48:46,309

 

1535
00:48:46,309 --> 00:48:49,030

 

1536
00:48:49,030 --> 00:48:51,430

 

1537
00:48:51,430 --> 00:48:54,470

 

1538
00:48:54,470 --> 00:48:57,030

 

1539
00:48:57,030 --> 00:48:59,829

 

1540
00:48:59,829 --> 00:48:59,839

 

1541
00:48:59,839 --> 00:49:02,470


1542
00:49:02,470 --> 00:49:03,990

 

1543
00:49:03,990 --> 00:49:06,630

 

1544
00:49:06,630 --> 00:49:09,349

 

1545
00:49:09,349 --> 00:49:12,309

 

1546
00:49:12,309 --> 00:49:15,270

 

1547
00:49:15,270 --> 00:49:17,430

 

1548
00:49:17,430 --> 00:49:21,750

 

1549
00:49:21,750 --> 00:49:23,190

 

1550
00:49:23,190 --> 00:49:24,790

 

1551
00:49:24,790 --> 00:49:27,829

 

1552
00:49:27,829 --> 00:49:29,750

 

1553
00:49:29,750 --> 00:49:31,670

 

1554
00:49:31,670 --> 00:49:34,230

 

1555
00:49:34,230 --> 00:49:34,240

 

1556
00:49:34,240 --> 00:49:36,549


1557
00:49:36,549 --> 00:49:38,470

 

1558
00:49:38,470 --> 00:49:40,470

 

1559
00:49:40,470 --> 00:49:42,069

 

1560
00:49:42,069 --> 00:49:46,950

 

1561
00:49:46,950 --> 00:49:49,430

 

1562
00:49:49,430 --> 00:49:51,750

 

1563
00:49:51,750 --> 00:49:54,549

 

1564
00:49:54,549 --> 00:49:56,309

 

1565
00:49:56,309 --> 00:49:58,549

 

1566
00:49:58,549 --> 00:50:00,230

 

1567
00:50:00,230 --> 00:50:00,240

 

1568
00:50:00,240 --> 00:50:01,270


1569
00:50:01,270 --> 00:50:02,470

 

1570
00:50:02,470 --> 00:50:02,480

 

1571
00:50:02,480 --> 00:50:04,150


1572
00:50:04,150 --> 00:50:04,160

 

1573
00:50:04,160 --> 00:50:04,870


1574
00:50:04,870 --> 00:50:07,030

 

1575
00:50:07,030 --> 00:50:07,910

 

1576
00:50:07,910 --> 00:50:10,630

 

1577
00:50:10,630 --> 00:50:12,390

 

1578
00:50:12,390 --> 00:50:14,549

 

1579
00:50:14,549 --> 00:50:15,670

 

1580
00:50:15,670 --> 00:50:17,349

 

1581
00:50:17,349 --> 00:50:18,870

 

1582
00:50:18,870 --> 00:50:18,880

 

1583
00:50:18,880 --> 00:50:19,750


1584
00:50:19,750 --> 00:50:20,950

 

1585
00:50:20,950 --> 00:50:23,430

 

1586
00:50:23,430 --> 00:50:25,270

 

1587
00:50:25,270 --> 00:50:27,510

 

1588
00:50:27,510 --> 00:50:29,109

 

1589
00:50:29,109 --> 00:50:30,470

 

1590
00:50:30,470 --> 00:50:31,829

 

1591
00:50:31,829 --> 00:50:33,510

 

1592
00:50:33,510 --> 00:50:36,630

 

1593
00:50:36,630 --> 00:50:36,640

 

1594
00:50:36,640 --> 00:50:38,870


1595
00:50:38,870 --> 00:50:41,750

 

1596
00:50:41,750 --> 00:50:41,760

 

1597
00:50:41,760 --> 00:50:43,750


1598
00:50:43,750 --> 00:50:45,670

 

1599
00:50:45,670 --> 00:50:49,270

 

1600
00:50:49,270 --> 00:50:52,069

 

1601
00:50:52,069 --> 00:50:54,390

 

1602
00:50:54,390 --> 00:50:55,510

 

1603
00:50:55,510 --> 00:50:57,270

 

1604
00:50:57,270 --> 00:51:00,230

 

1605
00:51:00,230 --> 00:51:04,630

 

1606
00:51:04,630 --> 00:51:07,270

 

1607
00:51:07,270 --> 00:51:08,790

 

1608
00:51:08,790 --> 00:51:10,549

 

1609
00:51:10,549 --> 00:51:10,559

 

1610
00:51:10,559 --> 00:51:11,040


1611
00:51:11,040 --> 00:51:11,050

 

1612
00:51:11,050 --> 00:51:15,829


1613
00:51:15,829 --> 00:51:18,069

 

1614
00:51:18,069 --> 00:51:20,549

 

1615
00:51:20,549 --> 00:51:22,710

 

1616
00:51:22,710 --> 00:51:29,190

 

1617
00:51:29,190 --> 00:51:29,200

 

1618
00:51:29,200 --> 00:51:30,950


1619
00:51:30,950 --> 00:51:32,870

 

1620
00:51:32,870 --> 00:51:35,750

 

1621
00:51:35,750 --> 00:51:37,829

 

1622
00:51:37,829 --> 00:51:40,390

 

1623
00:51:40,390 --> 00:51:44,069

 

1624
00:51:44,069 --> 00:51:46,470

 

1625
00:51:46,470 --> 00:51:48,630

 

1626
00:51:48,630 --> 00:51:50,630

 

1627
00:51:50,630 --> 00:51:52,390

 

1628
00:51:52,390 --> 00:51:54,710

 

1629
00:51:54,710 --> 00:51:57,589

 

1630
00:51:57,589 --> 00:51:58,620

 

1631
00:51:58,620 --> 00:51:58,630

 

1632
00:51:58,630 --> 00:52:01,950


1633
00:52:01,950 --> 00:52:05,510

 

1634
00:52:05,510 --> 00:52:08,069

 

1635
00:52:08,069 --> 00:52:09,349

 

1636
00:52:09,349 --> 00:52:12,069

 

1637
00:52:12,069 --> 00:52:14,390

 

1638
00:52:14,390 --> 00:52:15,990

 

1639
00:52:15,990 --> 00:52:20,309

 

1640
00:52:20,309 --> 00:52:22,549

 

1641
00:52:22,549 --> 00:52:24,390

 

1642
00:52:24,390 --> 00:52:27,349

 

1643
00:52:27,349 --> 00:52:29,430

 

1644
00:52:29,430 --> 00:52:32,069

 

1645
00:52:32,069 --> 00:52:33,990

 

1646
00:52:33,990 --> 00:52:35,670

 

1647
00:52:35,670 --> 00:52:35,680

 

1648
00:52:35,680 --> 00:52:36,710


1649
00:52:36,710 --> 00:52:36,720

 

1650
00:52:36,720 --> 00:52:37,430


1651
00:52:37,430 --> 00:52:40,230

 

1652
00:52:40,230 --> 00:52:42,069

 

1653
00:52:42,069 --> 00:52:42,079

 

1654
00:52:42,079 --> 00:52:43,190


1655
00:52:43,190 --> 00:52:46,710

 

1656
00:52:46,710 --> 00:52:48,470

 

1657
00:52:48,470 --> 00:52:51,510

 

1658
00:52:51,510 --> 00:52:53,030

 

1659
00:52:53,030 --> 00:52:55,510

 

1660
00:52:55,510 --> 00:52:57,910

 

1661
00:52:57,910 --> 00:53:00,150

 

1662
00:53:00,150 --> 00:53:02,710

 

1663
00:53:02,710 --> 00:53:04,390

 

1664
00:53:04,390 --> 00:53:05,829

 

1665
00:53:05,829 --> 00:53:09,030

 

1666
00:53:09,030 --> 00:53:10,150

 

1667
00:53:10,150 --> 00:53:11,829

 

1668
00:53:11,829 --> 00:53:11,839

 

1669
00:53:11,839 --> 00:53:14,390


1670
00:53:14,390 --> 00:53:16,470

 

1671
00:53:16,470 --> 00:53:16,480

 

1672
00:53:16,480 --> 00:53:18,390


1673
00:53:18,390 --> 00:53:21,430

 

1674
00:53:21,430 --> 00:53:23,430

 

1675
00:53:23,430 --> 00:53:26,069

 

1676
00:53:26,069 --> 00:53:29,750

 

1677
00:53:29,750 --> 00:53:30,790

 

1678
00:53:30,790 --> 00:53:32,950

 

1679
00:53:32,950 --> 00:53:34,870

 

1680
00:53:34,870 --> 00:53:37,270

 

1681
00:53:37,270 --> 00:53:38,630

 

1682
00:53:38,630 --> 00:53:40,630

 

1683
00:53:40,630 --> 00:53:43,670

 

1684
00:53:43,670 --> 00:53:46,230

 

1685
00:53:46,230 --> 00:53:47,990

 

1686
00:53:47,990 --> 00:53:50,150

 

1687
00:53:50,150 --> 00:53:51,990

 

1688
00:53:51,990 --> 00:53:54,870

 

1689
00:53:54,870 --> 00:53:57,270

 

1690
00:53:57,270 --> 00:54:01,190

 

1691
00:54:01,190 --> 00:54:01,200

 

1692
00:54:01,200 --> 00:54:03,190


1693
00:54:03,190 --> 00:54:04,630

 

1694
00:54:04,630 --> 00:54:05,829

 

1695
00:54:05,829 --> 00:54:08,150

 

1696
00:54:08,150 --> 00:54:12,870

 

1697
00:54:12,870 --> 00:54:14,390

 

1698
00:54:14,390 --> 00:54:17,510

 

1699
00:54:17,510 --> 00:54:19,349

 

1700
00:54:19,349 --> 00:54:22,390

 

1701
00:54:22,390 --> 00:54:24,230

 

1702
00:54:24,230 --> 00:54:28,230

 

1703
00:54:28,230 --> 00:54:30,950

 

1704
00:54:30,950 --> 00:54:33,589

 

1705
00:54:33,589 --> 00:54:36,309

 

1706
00:54:36,309 --> 00:54:40,230

 

1707
00:54:40,230 --> 00:54:43,349

 

1708
00:54:43,349 --> 00:54:44,870

 

1709
00:54:44,870 --> 00:54:50,549

 

1710
00:54:50,549 --> 00:54:50,559

 

1711
00:54:50,559 --> 00:54:53,270


1712
00:54:53,270 --> 00:54:55,349

 

1713
00:54:55,349 --> 00:54:56,789

 

1714
00:54:56,789 --> 00:54:58,390

 

1715
00:54:58,390 --> 00:55:00,470

 

1716
00:55:00,470 --> 00:55:01,910

 

1717
00:55:01,910 --> 00:55:03,829

 

1718
00:55:03,829 --> 00:55:03,839

 

1719
00:55:03,839 --> 00:55:05,589


1720
00:55:05,589 --> 00:55:07,910

 

1721
00:55:07,910 --> 00:55:07,920

 

1722
00:55:07,920 --> 00:55:09,190


1723
00:55:09,190 --> 00:55:10,309

 

1724
00:55:10,309 --> 00:55:12,630

 

1725
00:55:12,630 --> 00:55:15,510

 

1726
00:55:15,510 --> 00:55:15,520

 

1727
00:55:15,520 --> 00:55:17,030


1728
00:55:17,030 --> 00:55:20,789

 

1729
00:55:20,789 --> 00:55:23,510

 

1730
00:55:23,510 --> 00:55:27,589

 

1731
00:55:27,589 --> 00:55:29,990

 

1732
00:55:29,990 --> 00:55:31,589

 

1733
00:55:31,589 --> 00:55:33,430

 

1734
00:55:33,430 --> 00:55:35,349

 

1735
00:55:35,349 --> 00:55:38,630

 

1736
00:55:38,630 --> 00:55:41,829

 

1737
00:55:41,829 --> 00:55:43,750

 

1738
00:55:43,750 --> 00:55:43,760

 

1739
00:55:43,760 --> 00:55:45,670


1740
00:55:45,670 --> 00:55:47,270

 

1741
00:55:47,270 --> 00:55:48,630

 

1742
00:55:48,630 --> 00:55:53,030

 

1743
00:55:53,030 --> 00:55:56,710

 

1744
00:55:56,710 --> 00:55:59,349

 

1745
00:55:59,349 --> 00:56:00,829

 

1746
00:56:00,829 --> 00:56:02,870

 

1747
00:56:02,870 --> 00:56:05,510

 

1748
00:56:05,510 --> 00:56:07,430

 

1749
00:56:07,430 --> 00:56:09,750

 

1750
00:56:09,750 --> 00:56:12,549

 

1751
00:56:12,549 --> 00:56:14,710

 

1752
00:56:14,710 --> 00:56:16,870

 

1753
00:56:16,870 --> 00:56:19,190

 

1754
00:56:19,190 --> 00:56:20,309

 

1755
00:56:20,309 --> 00:56:23,829

 

1756
00:56:23,829 --> 00:56:25,670

 

1757
00:56:25,670 --> 00:56:27,190

 

1758
00:56:27,190 --> 00:56:29,910

 

1759
00:56:29,910 --> 00:56:31,990

 

1760
00:56:31,990 --> 00:56:34,069

 

1761
00:56:34,069 --> 00:56:34,079

 

1762
00:56:34,079 --> 00:56:35,349


1763
00:56:35,349 --> 00:56:37,510

 

1764
00:56:37,510 --> 00:56:37,520

 

1765
00:56:37,520 --> 00:56:38,470


1766
00:56:38,470 --> 00:56:39,589

 

1767
00:56:39,589 --> 00:56:41,109

 

1768
00:56:41,109 --> 00:56:43,270

 

1769
00:56:43,270 --> 00:56:45,510

 

1770
00:56:45,510 --> 00:56:47,990

 

1771
00:56:47,990 --> 00:56:49,349

 

1772
00:56:49,349 --> 00:56:51,270

 

1773
00:56:51,270 --> 00:56:53,109

 

1774
00:56:53,109 --> 00:56:56,150

 

1775
00:56:56,150 --> 00:56:57,990

 

1776
00:56:57,990 --> 00:57:00,230

 

1777
00:57:00,230 --> 00:57:02,950

 

1778
00:57:02,950 --> 00:57:04,309

 

1779
00:57:04,309 --> 00:57:06,710

 

1780
00:57:06,710 --> 00:57:10,309

 

1781
00:57:10,309 --> 00:57:11,910

 

1782
00:57:11,910 --> 00:57:14,150

 

1783
00:57:14,150 --> 00:57:15,750

 

1784
00:57:15,750 --> 00:57:17,589

 

1785
00:57:17,589 --> 00:57:20,390

 

1786
00:57:20,390 --> 00:57:22,470

 

1787
00:57:22,470 --> 00:57:23,910

 

1788
00:57:23,910 --> 00:57:25,670

 

1789
00:57:25,670 --> 00:57:26,470

 

1790
00:57:26,470 --> 00:57:28,230

 

1791
00:57:28,230 --> 00:57:30,630

 

1792
00:57:30,630 --> 00:57:33,510

 

1793
00:57:33,510 --> 00:57:35,270

 

1794
00:57:35,270 --> 00:57:38,789

 

1795
00:57:38,789 --> 00:57:40,309

 

1796
00:57:40,309 --> 00:57:41,750

 

1797
00:57:41,750 --> 00:57:43,589

 

1798
00:57:43,589 --> 00:57:46,150

 

1799
00:57:46,150 --> 00:57:46,160

 

1800
00:57:46,160 --> 00:57:47,030


1801
00:57:47,030 --> 00:57:48,789

 

1802
00:57:48,789 --> 00:57:48,799

 

1803
00:57:48,799 --> 00:57:49,990


1804
00:57:49,990 --> 00:57:51,589

 

1805
00:57:51,589 --> 00:57:53,430

 

1806
00:57:53,430 --> 00:57:55,109

 

1807
00:57:55,109 --> 00:57:56,630

 

1808
00:57:56,630 --> 00:57:58,789

 

1809
00:57:58,789 --> 00:58:01,589

 

1810
00:58:01,589 --> 00:58:01,599

 

1811
00:58:01,599 --> 00:58:02,789


1812
00:58:02,789 --> 00:58:04,950

 

1813
00:58:04,950 --> 00:58:06,789

 

1814
00:58:06,789 --> 00:58:08,789

 

1815
00:58:08,789 --> 00:58:10,069

 

1816
00:58:10,069 --> 00:58:11,510

 

1817
00:58:11,510 --> 00:58:13,270

 

1818
00:58:13,270 --> 00:58:15,670

 

1819
00:58:15,670 --> 00:58:16,870

 

1820
00:58:16,870 --> 00:58:18,390

 

1821
00:58:18,390 --> 00:58:19,910

 

1822
00:58:19,910 --> 00:58:22,950

 

1823
00:58:22,950 --> 00:58:24,870

 

1824
00:58:24,870 --> 00:58:26,549

 

1825
00:58:26,549 --> 00:58:28,870

 

1826
00:58:28,870 --> 00:58:30,309

 

1827
00:58:30,309 --> 00:58:31,589

 

1828
00:58:31,589 --> 00:58:32,630

 

1829
00:58:32,630 --> 00:58:34,390

 

1830
00:58:34,390 --> 00:58:38,230

 

1831
00:58:38,230 --> 00:58:38,240

 

1832
00:58:38,240 --> 00:58:40,710


1833
00:58:40,710 --> 00:58:42,309

 

1834
00:58:42,309 --> 00:58:45,670

 

1835
00:58:45,670 --> 00:58:46,870

 

1836
00:58:46,870 --> 00:58:48,549

 

1837
00:58:48,549 --> 00:58:51,910

 

1838
00:58:51,910 --> 00:58:54,150

 

1839
00:58:54,150 --> 00:58:56,549

 

1840
00:58:56,549 --> 00:58:58,630

 

1841
00:58:58,630 --> 00:59:01,109

 

1842
00:59:01,109 --> 00:59:04,390

 

1843
00:59:04,390 --> 00:59:07,510

 

1844
00:59:07,510 --> 00:59:10,230

 

1845
00:59:10,230 --> 00:59:10,240

 

1846
00:59:10,240 --> 00:59:11,030


1847
00:59:11,030 --> 00:59:12,230

 

1848
00:59:12,230 --> 00:59:13,910

 

1849
00:59:13,910 --> 00:59:18,069

 

1850
00:59:18,069 --> 00:59:20,950

 

1851
00:59:20,950 --> 00:59:23,510

 

1852
00:59:23,510 --> 00:59:25,270

 

1853
00:59:25,270 --> 00:59:28,069

 

1854
00:59:28,069 --> 00:59:29,510

 

1855
00:59:29,510 --> 00:59:34,549

 

1856
00:59:34,549 --> 00:59:34,559

 

1857
00:59:34,559 --> 00:59:36,390


1858
00:59:36,390 --> 00:59:39,270

 

1859
00:59:39,270 --> 00:59:41,109

 

1860
00:59:41,109 --> 00:59:43,270

 

1861
00:59:43,270 --> 00:59:45,270

 

1862
00:59:45,270 --> 00:59:47,589

 

1863
00:59:47,589 --> 00:59:48,789

 

1864
00:59:48,789 --> 00:59:50,230

 

1865
00:59:50,230 --> 00:59:52,069

 

1866
00:59:52,069 --> 00:59:54,470

 

1867
00:59:54,470 --> 00:59:56,309

 

1868
00:59:56,309 --> 00:59:58,870

 

1869
00:59:58,870 --> 01:00:01,030

 

1870
01:00:01,030 --> 01:00:02,789

 

1871
01:00:02,789 --> 01:00:05,430

 

1872
01:00:05,430 --> 01:00:07,030

 

1873
01:00:07,030 --> 01:00:09,670

 

1874
01:00:09,670 --> 01:00:09,680

 

1875
01:00:09,680 --> 01:00:10,630


1876
01:00:10,630 --> 01:00:12,470

 

1877
01:00:12,470 --> 01:00:14,630

 

1878
01:00:14,630 --> 01:00:16,630

 

1879
01:00:16,630 --> 01:00:18,789

 

1880
01:00:18,789 --> 01:00:22,150

 

1881
01:00:22,150 --> 01:00:23,829

 

1882
01:00:23,829 --> 01:00:25,109

 

1883
01:00:25,109 --> 01:00:25,119

 

1884
01:00:25,119 --> 01:00:26,309


1885
01:00:26,309 --> 01:00:30,789

 

1886
01:00:30,789 --> 01:00:32,230

 

1887
01:00:32,230 --> 01:00:35,670

 

1888
01:00:35,670 --> 01:00:36,950

 

1889
01:00:36,950 --> 01:00:39,190

 

1890
01:00:39,190 --> 01:00:40,549

 

1891
01:00:40,549 --> 01:00:43,030

 

1892
01:00:43,030 --> 01:00:44,710

 

1893
01:00:44,710 --> 01:00:44,720

 

1894
01:00:44,720 --> 01:00:45,670


1895
01:00:45,670 --> 01:00:47,430

 

1896
01:00:47,430 --> 01:00:50,150

 

1897
01:00:50,150 --> 01:00:52,789

 

1898
01:00:52,789 --> 01:00:54,630

 

1899
01:00:54,630 --> 01:00:57,589

 

1900
01:00:57,589 --> 01:00:59,030

 

1901
01:00:59,030 --> 01:01:01,349

 

1902
01:01:01,349 --> 01:01:03,430

 

1903
01:01:03,430 --> 01:01:05,430

 

1904
01:01:05,430 --> 01:01:08,950

 

1905
01:01:08,950 --> 01:01:10,470

 

1906
01:01:10,470 --> 01:01:11,750

 

1907
01:01:11,750 --> 01:01:11,760

 

1908
01:01:11,760 --> 01:01:12,549


1909
01:01:12,549 --> 01:01:12,559

 

1910
01:01:12,559 --> 01:01:13,829


1911
01:01:13,829 --> 01:01:16,150

 

1912
01:01:16,150 --> 01:01:19,030

 

1913
01:01:19,030 --> 01:01:20,710

 

1914
01:01:20,710 --> 01:01:20,720

 

1915
01:01:20,720 --> 01:01:22,069


1916
01:01:22,069 --> 01:01:23,829

 

1917
01:01:23,829 --> 01:01:27,910

 

1918
01:01:27,910 --> 01:01:29,750

 

1919
01:01:29,750 --> 01:01:32,069

 

1920
01:01:32,069 --> 01:01:33,990

 

1921
01:01:33,990 --> 01:01:34,000

 

1922
01:01:34,000 --> 01:01:35,910


1923
01:01:35,910 --> 01:01:39,030

 

1924
01:01:39,030 --> 01:01:42,230

 

1925
01:01:42,230 --> 01:01:44,069

 

1926
01:01:44,069 --> 01:01:44,079

 

1927
01:01:44,079 --> 01:01:45,510


1928
01:01:45,510 --> 01:01:45,520

 

1929
01:01:45,520 --> 01:01:46,470


1930
01:01:46,470 --> 01:01:48,150

 

1931
01:01:48,150 --> 01:01:49,750

 

1932
01:01:49,750 --> 01:01:51,109

 

1933
01:01:51,109 --> 01:01:52,789

 

1934
01:01:52,789 --> 01:01:55,589

 

1935
01:01:55,589 --> 01:01:58,150

 

1936
01:01:58,150 --> 01:01:59,750

 

1937
01:01:59,750 --> 01:02:00,789

 

1938
01:02:00,789 --> 01:02:03,029

 

1939
01:02:03,029 --> 01:02:06,390

 

1940
01:02:06,390 --> 01:02:08,630

 

1941
01:02:08,630 --> 01:02:12,069

 

1942
01:02:12,069 --> 01:02:13,829

 

1943
01:02:13,829 --> 01:02:15,990

 

1944
01:02:15,990 --> 01:02:16,870

 

1945
01:02:16,870 --> 01:02:18,950

 

1946
01:02:18,950 --> 01:02:22,150

 

1947
01:02:22,150 --> 01:02:24,789

 

1948
01:02:24,789 --> 01:02:27,270

 

1949
01:02:27,270 --> 01:02:30,710

 

1950
01:02:30,710 --> 01:02:32,069

 

1951
01:02:32,069 --> 01:02:34,710

 

1952
01:02:34,710 --> 01:02:35,829

 

1953
01:02:35,829 --> 01:02:38,230

 

1954
01:02:38,230 --> 01:02:40,390

 

1955
01:02:40,390 --> 01:02:43,750

 

1956
01:02:43,750 --> 01:02:46,309

 

1957
01:02:46,309 --> 01:02:48,390

 

1958
01:02:48,390 --> 01:02:49,750

 

1959
01:02:49,750 --> 01:02:52,309

 

1960
01:02:52,309 --> 01:02:55,029

 

1961
01:02:55,029 --> 01:02:57,430

 

1962
01:02:57,430 --> 01:02:57,440

 

1963
01:02:57,440 --> 01:02:59,029


1964
01:02:59,029 --> 01:03:00,789

 

1965
01:03:00,789 --> 01:03:03,190

 

1966
01:03:03,190 --> 01:03:05,750

 

1967
01:03:05,750 --> 01:03:08,390

 

1968
01:03:08,390 --> 01:03:10,309

 

1969
01:03:10,309 --> 01:03:11,750

 

1970
01:03:11,750 --> 01:03:14,230

 

1971
01:03:14,230 --> 01:03:16,870

 

1972
01:03:16,870 --> 01:03:18,150

 

1973
01:03:18,150 --> 01:03:20,069

 

1974
01:03:20,069 --> 01:03:21,990

 

1975
01:03:21,990 --> 01:03:24,069

 

1976
01:03:24,069 --> 01:03:25,430

 

1977
01:03:25,430 --> 01:03:27,589

 

1978
01:03:27,589 --> 01:03:29,750

 

1979
01:03:29,750 --> 01:03:31,670

 

1980
01:03:31,670 --> 01:03:31,680

 

1981
01:03:31,680 --> 01:03:33,910


1982
01:03:33,910 --> 01:03:36,230

 

1983
01:03:36,230 --> 01:03:37,750

 

1984
01:03:37,750 --> 01:03:37,760

 

1985
01:03:37,760 --> 01:03:39,430


1986
01:03:39,430 --> 01:03:39,440

 

1987
01:03:39,440 --> 01:03:41,430


1988
01:03:41,430 --> 01:03:43,750

 

1989
01:03:43,750 --> 01:03:45,829

 

1990
01:03:45,829 --> 01:03:47,829

 

1991
01:03:47,829 --> 01:03:49,829

 

1992
01:03:49,829 --> 01:03:52,309

 

1993
01:03:52,309 --> 01:03:52,319

 

1994
01:03:52,319 --> 01:03:53,829


1995
01:03:53,829 --> 01:03:56,069

 

1996
01:03:56,069 --> 01:03:57,589

 

1997
01:03:57,589 --> 01:04:02,950

 

1998
01:04:02,950 --> 01:04:05,510

 

1999
01:04:05,510 --> 01:04:09,029

 

2000
01:04:09,029 --> 01:04:11,430

 

2001
01:04:11,430 --> 01:04:14,230

 

2002
01:04:14,230 --> 01:04:17,029

 

2003
01:04:17,029 --> 01:04:18,950

 

2004
01:04:18,950 --> 01:04:20,710

 

2005
01:04:20,710 --> 01:04:23,750

 

2006
01:04:23,750 --> 01:04:26,470

 

2007
01:04:26,470 --> 01:04:28,470

 

2008
01:04:28,470 --> 01:04:30,789

 

2009
01:04:30,789 --> 01:04:33,349

 

2010
01:04:33,349 --> 01:04:36,950

 

2011
01:04:36,950 --> 01:04:38,870

 

2012
01:04:38,870 --> 01:04:41,829

 

2013
01:04:41,829 --> 01:04:45,750

 

2014
01:04:45,750 --> 01:04:47,910

 

2015
01:04:47,910 --> 01:04:49,829

 

2016
01:04:49,829 --> 01:04:51,589

 

2017
01:04:51,589 --> 01:04:52,950

 

2018
01:04:52,950 --> 01:04:54,549

 

2019
01:04:54,549 --> 01:04:56,710

 

2020
01:04:56,710 --> 01:04:59,190

 

2021
01:04:59,190 --> 01:05:02,150

 

2022
01:05:02,150 --> 01:05:04,069

 

2023
01:05:04,069 --> 01:05:05,750

 

2024
01:05:05,750 --> 01:05:07,430

 

2025
01:05:07,430 --> 01:05:09,510

 

2026
01:05:09,510 --> 01:05:12,630

 

2027
01:05:12,630 --> 01:05:14,230

 

2028
01:05:14,230 --> 01:05:15,910

 

2029
01:05:15,910 --> 01:05:17,589

 

2030
01:05:17,589 --> 01:05:17,599

 

2031
01:05:17,599 --> 01:05:19,349


2032
01:05:19,349 --> 01:05:23,029

 

2033
01:05:23,029 --> 01:05:26,150

 

2034
01:05:26,150 --> 01:05:26,160

 

2035
01:05:26,160 --> 01:05:27,109


2036
01:05:27,109 --> 01:05:28,630

 

2037
01:05:28,630 --> 01:05:28,640

 

2038
01:05:28,640 --> 01:05:29,510


2039
01:05:29,510 --> 01:05:30,950

 

2040
01:05:30,950 --> 01:05:33,430

 

2041
01:05:33,430 --> 01:05:36,549

 

2042
01:05:36,549 --> 01:05:38,789

 

2043
01:05:38,789 --> 01:05:40,390

 

2044
01:05:40,390 --> 01:05:42,950

 

2045
01:05:42,950 --> 01:05:46,069

 

2046
01:05:46,069 --> 01:05:48,870

 

2047
01:05:48,870 --> 01:05:50,789

 

2048
01:05:50,789 --> 01:05:50,799

 

2049
01:05:50,799 --> 01:05:52,470


2050
01:05:52,470 --> 01:05:53,589

 

2051
01:05:53,589 --> 01:05:56,950

 

2052
01:05:56,950 --> 01:05:56,960

 

2053
01:05:56,960 --> 01:05:57,990


2054
01:05:57,990 --> 01:05:59,990

 

2055
01:05:59,990 --> 01:06:01,589

 

2056
01:06:01,589 --> 01:06:04,309

 

2057
01:06:04,309 --> 01:06:06,390

 

2058
01:06:06,390 --> 01:06:08,470

 

2059
01:06:08,470 --> 01:06:10,390

 

2060
01:06:10,390 --> 01:06:11,829

 

2061
01:06:11,829 --> 01:06:13,670

 

2062
01:06:13,670 --> 01:06:16,309

 

2063
01:06:16,309 --> 01:06:17,430

 

2064
01:06:17,430 --> 01:06:19,190

 

2065
01:06:19,190 --> 01:06:21,029

 

2066
01:06:21,029 --> 01:06:23,990

 

2067
01:06:23,990 --> 01:06:26,470

 

2068
01:06:26,470 --> 01:06:28,309

 

2069
01:06:28,309 --> 01:06:31,109

 

2070
01:06:31,109 --> 01:06:32,549

 

2071
01:06:32,549 --> 01:06:34,549

 

2072
01:06:34,549 --> 01:06:36,870

 

2073
01:06:36,870 --> 01:06:39,510

 

2074
01:06:39,510 --> 01:06:40,870

 

2075
01:06:40,870 --> 01:06:42,390

 

2076
01:06:42,390 --> 01:06:42,400

 

2077
01:06:42,400 --> 01:06:43,349


2078
01:06:43,349 --> 01:06:50,150

 

2079
01:06:50,150 --> 01:06:54,950

 

2080
01:06:54,950 --> 01:07:00,789

 

2081
01:07:00,789 --> 01:07:05,069

 

2082
01:07:05,069 --> 01:07:07,910

 

2083
01:07:07,910 --> 01:07:09,910

 

2084
01:07:09,910 --> 01:07:11,190

 

2085
01:07:11,190 --> 01:07:13,510

 

2086
01:07:13,510 --> 01:07:14,870

 

2087
01:07:14,870 --> 01:07:16,870

 

2088
01:07:16,870 --> 01:07:18,870

 

2089
01:07:18,870 --> 01:07:18,880

 

2090
01:07:18,880 --> 01:07:19,910


2091
01:07:19,910 --> 01:07:22,230

 

2092
01:07:22,230 --> 01:07:24,950

 

2093
01:07:24,950 --> 01:07:26,630

 

2094
01:07:26,630 --> 01:07:28,470

 

2095
01:07:28,470 --> 01:07:29,910

 

2096
01:07:29,910 --> 01:07:31,270

 

2097
01:07:31,270 --> 01:07:31,280

 

2098
01:07:31,280 --> 01:07:32,870


2099
01:07:32,870 --> 01:07:32,880

 

2100
01:07:32,880 --> 01:07:34,470


2101
01:07:34,470 --> 01:07:38,390

 

2102
01:07:38,390 --> 01:07:40,230

 

2103
01:07:40,230 --> 01:07:45,109

 

2104
01:07:45,109 --> 01:07:47,990

 

2105
01:07:47,990 --> 01:07:49,349

 

2106
01:07:49,349 --> 01:07:51,430

 

2107
01:07:51,430 --> 01:07:53,349

 

2108
01:07:53,349 --> 01:07:56,710

 

2109
01:07:56,710 --> 01:07:59,349

 

2110
01:07:59,349 --> 01:07:59,359

 

2111
01:07:59,359 --> 01:08:01,349


2112
01:08:01,349 --> 01:08:03,270

 

2113
01:08:03,270 --> 01:08:04,950

 

2114
01:08:04,950 --> 01:08:07,430

 

2115
01:08:07,430 --> 01:08:07,440

 

2116
01:08:07,440 --> 01:08:08,309


2117
01:08:08,309 --> 01:08:09,829

 

2118
01:08:09,829 --> 01:08:11,510

 

2119
01:08:11,510 --> 01:08:13,430

 

2120
01:08:13,430 --> 01:08:13,440

 

2121
01:08:13,440 --> 01:08:14,390


2122
01:08:14,390 --> 01:08:17,349

 

2123
01:08:17,349 --> 01:08:19,590

 

2124
01:08:19,590 --> 01:08:22,550

 

2125
01:08:22,550 --> 01:08:24,070

 

2126
01:08:24,070 --> 01:08:26,709

 

2127
01:08:26,709 --> 01:08:28,550

 

2128
01:08:28,550 --> 01:08:31,189

 

2129
01:08:31,189 --> 01:08:32,390

 

2130
01:08:32,390 --> 01:08:35,749

 

2131
01:08:35,749 --> 01:08:38,229

 

2132
01:08:38,229 --> 01:08:38,239

 

2133
01:08:38,239 --> 01:08:39,749


2134
01:08:39,749 --> 01:08:41,910

 

2135
01:08:41,910 --> 01:08:41,920

 

2136
01:08:41,920 --> 01:08:43,269


2137
01:08:43,269 --> 01:08:45,910

 

2138
01:08:45,910 --> 01:08:47,349

 

2139
01:08:47,349 --> 01:08:49,349

 

2140
01:08:49,349 --> 01:08:51,269

 

2141
01:08:51,269 --> 01:08:53,030

 

2142
01:08:53,030 --> 01:08:56,950

 

2143
01:08:56,950 --> 01:08:57,910

 

2144
01:08:57,910 --> 01:09:00,630

 

2145
01:09:00,630 --> 01:09:02,950

 

2146
01:09:02,950 --> 01:09:05,990

 

2147
01:09:05,990 --> 01:09:10,070

 

2148
01:09:10,070 --> 01:09:12,390

 

2149
01:09:12,390 --> 01:09:14,789

 

2150
01:09:14,789 --> 01:09:19,269

 

2151
01:09:19,269 --> 01:09:21,510

 

2152
01:09:21,510 --> 01:09:24,390

 

2153
01:09:24,390 --> 01:09:25,910

 

2154
01:09:25,910 --> 01:09:27,510

 

2155
01:09:27,510 --> 01:09:31,030

 

2156
01:09:31,030 --> 01:09:32,709

 

2157
01:09:32,709 --> 01:09:35,990

 

2158
01:09:35,990 --> 01:09:37,990

 

2159
01:09:37,990 --> 01:09:39,910

 

2160
01:09:39,910 --> 01:09:41,990

 

2161
01:09:41,990 --> 01:09:43,189

 

2162
01:09:43,189 --> 01:09:44,789

 

2163
01:09:44,789 --> 01:09:47,110

 

2164
01:09:47,110 --> 01:09:49,269

 

2165
01:09:49,269 --> 01:09:50,789

 

2166
01:09:50,789 --> 01:09:52,789

 

2167
01:09:52,789 --> 01:09:55,510

 

2168
01:09:55,510 --> 01:09:57,750

 

2169
01:09:57,750 --> 01:09:59,510

 

2170
01:09:59,510 --> 01:10:02,950

 

2171
01:10:02,950 --> 01:10:05,590

 

2172
01:10:05,590 --> 01:10:05,600

 

2173
01:10:05,600 --> 01:10:07,910


2174
01:10:07,910 --> 01:10:10,790

 

2175
01:10:10,790 --> 01:10:12,550

 

2176
01:10:12,550 --> 01:10:15,830

 

2177
01:10:15,830 --> 01:10:18,550

 

2178
01:10:18,550 --> 01:10:21,270

 

2179
01:10:21,270 --> 01:10:23,430

 

2180
01:10:23,430 --> 01:10:25,189

 

2181
01:10:25,189 --> 01:10:27,189

 

2182
01:10:27,189 --> 01:10:27,199

 

2183
01:10:27,199 --> 01:10:29,430


2184
01:10:29,430 --> 01:10:29,440

 

2185
01:10:29,440 --> 01:10:31,189


2186
01:10:31,189 --> 01:10:33,189

 

2187
01:10:33,189 --> 01:10:35,669

 

2188
01:10:35,669 --> 01:10:35,679

 

2189
01:10:35,679 --> 01:10:36,470


2190
01:10:36,470 --> 01:10:38,229

 

2191
01:10:38,229 --> 01:10:40,070

 

2192
01:10:40,070 --> 01:10:42,709

 

2193
01:10:42,709 --> 01:10:45,350

 

2194
01:10:45,350 --> 01:10:47,990

 

2195
01:10:47,990 --> 01:10:49,350

 

2196
01:10:49,350 --> 01:10:52,470

 

2197
01:10:52,470 --> 01:10:55,030

 

2198
01:10:55,030 --> 01:10:55,040

 

2199
01:10:55,040 --> 01:10:58,950


2200
01:10:58,950 --> 01:11:02,149

 

2201
01:11:02,149 --> 01:11:02,159

 

2202
01:11:02,159 --> 01:11:03,430


2203
01:11:03,430 --> 01:11:05,270

 

2204
01:11:05,270 --> 01:11:05,280

 

2205
01:11:05,280 --> 01:11:06,310


2206
01:11:06,310 --> 01:11:08,870

 

2207
01:11:08,870 --> 01:11:10,790

 

2208
01:11:10,790 --> 01:11:10,800

 

2209
01:11:10,800 --> 01:11:12,790


2210
01:11:12,790 --> 01:11:14,470

 

2211
01:11:14,470 --> 01:11:16,390

 

2212
01:11:16,390 --> 01:11:19,110

 

2213
01:11:19,110 --> 01:11:22,790

 

2214
01:11:22,790 --> 01:11:24,709

 

2215
01:11:24,709 --> 01:11:27,750

 

2216
01:11:27,750 --> 01:11:29,590

 

2217
01:11:29,590 --> 01:11:31,189

 

2218
01:11:31,189 --> 01:11:33,430

 

2219
01:11:33,430 --> 01:11:34,870

 

2220
01:11:34,870 --> 01:11:38,870

 

2221
01:11:38,870 --> 01:11:40,790

 

2222
01:11:40,790 --> 01:11:42,790

 

2223
01:11:42,790 --> 01:11:42,800

 

2224
01:11:42,800 --> 01:11:44,229


2225
01:11:44,229 --> 01:11:46,870

 

2226
01:11:46,870 --> 01:11:49,830

 

2227
01:11:49,830 --> 01:11:54,550

 

2228
01:11:54,550 --> 01:11:56,229

 

2229
01:11:56,229 --> 01:11:57,350

 

2230
01:11:57,350 --> 01:11:59,110

 

2231
01:11:59,110 --> 01:12:00,950

 

2232
01:12:00,950 --> 01:12:03,830

 

2233
01:12:03,830 --> 01:12:06,310

 

2234
01:12:06,310 --> 01:12:07,990

 

2235
01:12:07,990 --> 01:12:10,310

 

2236
01:12:10,310 --> 01:12:12,470

 

2237
01:12:12,470 --> 01:12:13,590

 

2238
01:12:13,590 --> 01:12:15,910

 

2239
01:12:15,910 --> 01:12:18,070

 

2240
01:12:18,070 --> 01:12:19,430

 

2241
01:12:19,430 --> 01:12:19,440

 

2242
01:12:19,440 --> 01:12:21,189


2243
01:12:21,189 --> 01:12:22,790

 

2244
01:12:22,790 --> 01:12:22,800

 

2245
01:12:22,800 --> 01:12:24,870


2246
01:12:24,870 --> 01:12:27,910

 

2247
01:12:27,910 --> 01:12:27,920

 

2248
01:12:27,920 --> 01:12:29,990


2249
01:12:29,990 --> 01:12:31,430

 

2250
01:12:31,430 --> 01:12:34,149

 

2251
01:12:34,149 --> 01:12:37,830

 

2252
01:12:37,830 --> 01:12:40,229

 

2253
01:12:40,229 --> 01:12:40,239

 

2254
01:12:40,239 --> 01:12:41,910


2255
01:12:41,910 --> 01:12:43,270

 

2256
01:12:43,270 --> 01:12:45,270

 

2257
01:12:45,270 --> 01:12:47,510

 

2258
01:12:47,510 --> 01:12:49,350

 

2259
01:12:49,350 --> 01:12:51,510

 

2260
01:12:51,510 --> 01:12:54,070

 

2261
01:12:54,070 --> 01:12:54,080

 

2262
01:12:54,080 --> 01:12:55,189


2263
01:12:55,189 --> 01:12:59,030

 

2264
01:12:59,030 --> 01:12:59,040

 

2265
01:12:59,040 --> 01:13:00,149


2266
01:13:00,149 --> 01:13:01,430

 

2267
01:13:01,430 --> 01:13:03,110

 

2268
01:13:03,110 --> 01:13:04,870

 

2269
01:13:04,870 --> 01:13:07,990

 

2270
01:13:07,990 --> 01:13:11,270

 

2271
01:13:11,270 --> 01:13:13,510

 

2272
01:13:13,510 --> 01:13:15,750

 

2273
01:13:15,750 --> 01:13:16,870

 

2274
01:13:16,870 --> 01:13:19,669

 

2275
01:13:19,669 --> 01:13:21,510

 

2276
01:13:21,510 --> 01:13:23,750

 

2277
01:13:23,750 --> 01:13:25,990

 

2278
01:13:25,990 --> 01:13:27,350

 

2279
01:13:27,350 --> 01:13:29,669

 

2280
01:13:29,669 --> 01:13:31,750

 

2281
01:13:31,750 --> 01:13:33,990

 

2282
01:13:33,990 --> 01:13:36,310

 

2283
01:13:36,310 --> 01:13:36,320

 

2284
01:13:36,320 --> 01:13:37,270


2285
01:13:37,270 --> 01:13:37,280

 

2286
01:13:37,280 --> 01:13:38,149


2287
01:13:38,149 --> 01:13:40,310

 

2288
01:13:40,310 --> 01:13:41,990

 

2289
01:13:41,990 --> 01:13:44,870

 

2290
01:13:44,870 --> 01:13:46,709

 

2291
01:13:46,709 --> 01:13:49,030

 

2292
01:13:49,030 --> 01:13:50,870

 

2293
01:13:50,870 --> 01:13:52,870

 

2294
01:13:52,870 --> 01:13:54,390

 

2295
01:13:54,390 --> 01:13:54,400

 

2296
01:13:54,400 --> 01:13:55,669


2297
01:13:55,669 --> 01:13:57,669

 

2298
01:13:57,669 --> 01:13:59,189

 

2299
01:13:59,189 --> 01:14:01,430

 

2300
01:14:01,430 --> 01:14:03,910

 

2301
01:14:03,910 --> 01:14:05,910

 

2302
01:14:05,910 --> 01:14:07,430

 

2303
01:14:07,430 --> 01:14:10,229

 

2304
01:14:10,229 --> 01:14:12,310

 

2305
01:14:12,310 --> 01:14:14,550

 

2306
01:14:14,550 --> 01:14:16,070

 

2307
01:14:16,070 --> 01:14:16,080

 

2308
01:14:16,080 --> 01:14:16,460


2309
01:14:16,460 --> 01:14:16,470

 

2310
01:14:16,470 --> 01:14:18,149


2311
01:14:18,149 --> 01:14:20,550

 

2312
01:14:20,550 --> 01:14:22,550

 

2313
01:14:22,550 --> 01:14:25,030

 

2314
01:14:25,030 --> 01:14:27,430

 

2315
01:14:27,430 --> 01:14:27,440

 

2316
01:14:27,440 --> 01:14:28,630


2317
01:14:28,630 --> 01:14:28,640

 

2318
01:14:28,640 --> 01:14:29,669


2319
01:14:29,669 --> 01:14:31,669

 

2320
01:14:31,669 --> 01:14:33,910

 

2321
01:14:33,910 --> 01:14:33,920

 

2322
01:14:33,920 --> 01:14:34,790


2323
01:14:34,790 --> 01:14:37,590

 

2324
01:14:37,590 --> 01:14:40,550

 

2325
01:14:40,550 --> 01:14:42,630

 

2326
01:14:42,630 --> 01:14:44,870

 

2327
01:14:44,870 --> 01:14:46,550

 

2328
01:14:46,550 --> 01:14:46,560

 

2329
01:14:46,560 --> 01:14:47,669


2330
01:14:47,669 --> 01:14:50,229

 

2331
01:14:50,229 --> 01:14:51,510

 

2332
01:14:51,510 --> 01:14:53,990

 

2333
01:14:53,990 --> 01:14:56,950

 

2334
01:14:56,950 --> 01:14:59,350

 

2335
01:14:59,350 --> 01:15:01,510

 

2336
01:15:01,510 --> 01:15:02,390

 

2337
01:15:02,390 --> 01:15:04,149

 

2338
01:15:04,149 --> 01:15:05,510

 

2339
01:15:05,510 --> 01:15:06,630

 

2340
01:15:06,630 --> 01:15:08,550

 

2341
01:15:08,550 --> 01:15:10,149

 

2342
01:15:10,149 --> 01:15:12,790

 

2343
01:15:12,790 --> 01:15:15,590

 

2344
01:15:15,590 --> 01:15:17,750

 

2345
01:15:17,750 --> 01:15:19,350

 

2346
01:15:19,350 --> 01:15:21,270

 

2347
01:15:21,270 --> 01:15:23,510

 

2348
01:15:23,510 --> 01:15:24,870

 

2349
01:15:24,870 --> 01:15:26,310

 

2350
01:15:26,310 --> 01:15:27,910

 

2351
01:15:27,910 --> 01:15:29,910

 

2352
01:15:29,910 --> 01:15:33,030

 

2353
01:15:33,030 --> 01:15:33,040

 

2354
01:15:33,040 --> 01:15:34,470


2355
01:15:34,470 --> 01:15:36,470

 

2356
01:15:36,470 --> 01:15:38,870

 

2357
01:15:38,870 --> 01:15:40,870

 

2358
01:15:40,870 --> 01:15:43,750

 

2359
01:15:43,750 --> 01:15:46,390

 

2360
01:15:46,390 --> 01:15:47,669

 

2361
01:15:47,669 --> 01:15:51,110

 

2362
01:15:51,110 --> 01:15:53,030

 

2363
01:15:53,030 --> 01:15:55,189

 

2364
01:15:55,189 --> 01:15:57,270

 

2365
01:15:57,270 --> 01:15:57,280

 

2366
01:15:57,280 --> 01:15:58,229


2367
01:15:58,229 --> 01:16:00,390

 

2368
01:16:00,390 --> 01:16:02,310

 

2369
01:16:02,310 --> 01:16:02,320

 

2370
01:16:02,320 --> 01:16:03,189


2371
01:16:03,189 --> 01:16:04,630

 

2372
01:16:04,630 --> 01:16:07,110

 

2373
01:16:07,110 --> 01:16:09,830

 

2374
01:16:09,830 --> 01:16:12,790

 

2375
01:16:12,790 --> 01:16:15,189

 

2376
01:16:15,189 --> 01:16:18,310

 

2377
01:16:18,310 --> 01:16:20,149

 

2378
01:16:20,149 --> 01:16:21,750

 

2379
01:16:21,750 --> 01:16:21,760

 

2380
01:16:21,760 --> 01:16:24,149


2381
01:16:24,149 --> 01:16:25,830

 

2382
01:16:25,830 --> 01:16:27,750

 

2383
01:16:27,750 --> 01:16:27,760

 

2384
01:16:27,760 --> 01:16:29,189


2385
01:16:29,189 --> 01:16:29,199

 

2386
01:16:29,199 --> 01:16:30,310


2387
01:16:30,310 --> 01:16:32,390

 

2388
01:16:32,390 --> 01:16:35,350

 

2389
01:16:35,350 --> 01:16:39,590

 

2390
01:16:39,590 --> 01:16:42,709

 

2391
01:16:42,709 --> 01:16:44,470

 

2392
01:16:44,470 --> 01:16:44,480

 

2393
01:16:44,480 --> 01:16:45,830


2394
01:16:45,830 --> 01:16:52,470

 

2395
01:16:52,470 --> 01:16:52,480

 

2396
01:16:52,480 --> 01:16:53,669


2397
01:16:53,669 --> 01:16:56,390

 

2398
01:16:56,390 --> 01:16:58,790

 

2399
01:16:58,790 --> 01:17:00,790

 

2400
01:17:00,790 --> 01:17:03,270

 

2401
01:17:03,270 --> 01:17:06,310

 

2402
01:17:06,310 --> 01:17:07,830

 

2403
01:17:07,830 --> 01:17:07,840

 

2404
01:17:07,840 --> 01:17:08,870


2405
01:17:08,870 --> 01:17:10,390

 

2406
01:17:10,390 --> 01:17:10,400

 

2407
01:17:10,400 --> 01:17:11,430


2408
01:17:11,430 --> 01:17:13,510

 

2409
01:17:13,510 --> 01:17:13,520

 

2410
01:17:13,520 --> 01:17:15,430


2411
01:17:15,430 --> 01:17:16,709

 

2412
01:17:16,709 --> 01:17:19,990

 

2413
01:17:19,990 --> 01:17:20,000

 

2414
01:17:20,000 --> 01:17:21,030


2415
01:17:21,030 --> 01:17:23,110

 

2416
01:17:23,110 --> 01:17:23,120

 

2417
01:17:23,120 --> 01:17:24,229


2418
01:17:24,229 --> 01:17:28,310

 

2419
01:17:28,310 --> 01:17:30,950

 

2420
01:17:30,950 --> 01:17:32,470

 

2421
01:17:32,470 --> 01:17:34,310

 

2422
01:17:34,310 --> 01:17:36,229

 

2423
01:17:36,229 --> 01:17:38,229

 

2424
01:17:38,229 --> 01:17:40,070

 

2425
01:17:40,070 --> 01:17:41,910

 

2426
01:17:41,910 --> 01:17:44,470

 

2427
01:17:44,470 --> 01:17:46,310

 

2428
01:17:46,310 --> 01:17:48,950

 

2429
01:17:48,950 --> 01:17:48,960

 

2430
01:17:48,960 --> 01:17:49,990


2431
01:17:49,990 --> 01:17:52,550

 

2432
01:17:52,550 --> 01:17:55,430

 

2433
01:17:55,430 --> 01:17:56,830

 

2434
01:17:56,830 --> 01:17:59,590

 

2435
01:17:59,590 --> 01:18:03,110

 

2436
01:18:03,110 --> 01:18:03,120

 

2437
01:18:03,120 --> 01:18:04,149


2438
01:18:04,149 --> 01:18:04,159

 

2439
01:18:04,159 --> 01:18:06,229


2440
01:18:06,229 --> 01:18:09,350

 

2441
01:18:09,350 --> 01:18:11,430

 

2442
01:18:11,430 --> 01:18:14,390

 

2443
01:18:14,390 --> 01:18:16,790

 

2444
01:18:16,790 --> 01:18:18,709

 

2445
01:18:18,709 --> 01:18:19,990

 

2446
01:18:19,990 --> 01:18:22,709

 

2447
01:18:22,709 --> 01:18:24,470

 

2448
01:18:24,470 --> 01:18:26,870

 

2449
01:18:26,870 --> 01:18:26,880

 

2450
01:18:26,880 --> 01:18:28,310


2451
01:18:28,310 --> 01:18:28,320

 

2452
01:18:28,320 --> 01:18:30,709


2453
01:18:30,709 --> 01:18:32,709

 

2454
01:18:32,709 --> 01:18:34,390

 

2455
01:18:34,390 --> 01:18:36,470

 

2456
01:18:36,470 --> 01:18:37,990

 

2457
01:18:37,990 --> 01:18:38,000

 

2458
01:18:38,000 --> 01:18:39,510


2459
01:18:39,510 --> 01:18:42,550

 

2460
01:18:42,550 --> 01:18:44,709

 

2461
01:18:44,709 --> 01:18:46,229

 

2462
01:18:46,229 --> 01:18:47,990

 

2463
01:18:47,990 --> 01:18:49,669

 

2464
01:18:49,669 --> 01:18:52,070

 

2465
01:18:52,070 --> 01:18:52,080

 

2466
01:18:52,080 --> 01:18:53,030


2467
01:18:53,030 --> 01:18:54,470

 

2468
01:18:54,470 --> 01:18:55,910

 

2469
01:18:55,910 --> 01:18:57,510

 

2470
01:18:57,510 --> 01:18:58,870

 

2471
01:18:58,870 --> 01:18:58,880

 

2472
01:18:58,880 --> 01:18:59,910


2473
01:18:59,910 --> 01:19:01,189

 

2474
01:19:01,189 --> 01:19:01,199

 

2475
01:19:01,199 --> 01:19:02,149


2476
01:19:02,149 --> 01:19:02,159

 

2477
01:19:02,159 --> 01:19:03,430


2478
01:19:03,430 --> 01:19:05,669

 

2479
01:19:05,669 --> 01:19:08,149

 

2480
01:19:08,149 --> 01:19:09,510

 

2481
01:19:09,510 --> 01:19:11,910

 

2482
01:19:11,910 --> 01:19:13,430

 

2483
01:19:13,430 --> 01:19:15,350

 

2484
01:19:15,350 --> 01:19:17,669

 

2485
01:19:17,669 --> 01:19:19,510

 

2486
01:19:19,510 --> 01:19:19,520

 

2487
01:19:19,520 --> 01:19:20,550


2488
01:19:20,550 --> 01:19:22,310

 

2489
01:19:22,310 --> 01:19:24,229

 

2490
01:19:24,229 --> 01:19:25,510

 

2491
01:19:25,510 --> 01:19:28,070

 

2492
01:19:28,070 --> 01:19:30,229

 

2493
01:19:30,229 --> 01:19:33,990

 

2494
01:19:33,990 --> 01:19:37,030

 

2495
01:19:37,030 --> 01:19:38,709

 

2496
01:19:38,709 --> 01:19:43,189

 

2497
01:19:43,189 --> 01:19:44,630

 

2498
01:19:44,630 --> 01:19:46,070

 

2499
01:19:46,070 --> 01:19:47,669

 

2500
01:19:47,669 --> 01:19:51,270

 

2501
01:19:51,270 --> 01:19:53,030

 

2502
01:19:53,030 --> 01:19:57,910

 

2503
01:19:57,910 --> 01:20:01,669

 

2504
01:20:01,669 --> 01:20:03,030

 

2505
01:20:03,030 --> 01:20:05,910

 

2506
01:20:05,910 --> 01:20:05,920

 

2507
01:20:05,920 --> 01:20:06,870


2508
01:20:06,870 --> 01:20:09,270

 

2509
01:20:09,270 --> 01:20:10,709

 

2510
01:20:10,709 --> 01:20:10,719

 

2511
01:20:10,719 --> 01:20:11,830


2512
01:20:11,830 --> 01:20:13,590

 

2513
01:20:13,590 --> 01:20:15,750

 

2514
01:20:15,750 --> 01:20:17,350

 

2515
01:20:17,350 --> 01:20:18,790

 

2516
01:20:18,790 --> 01:20:21,189

 

2517
01:20:21,189 --> 01:20:22,950

 

2518
01:20:22,950 --> 01:20:24,310

 

2519
01:20:24,310 --> 01:20:25,669

 

2520
01:20:25,669 --> 01:20:27,189

 

2521
01:20:27,189 --> 01:20:29,030

 

2522
01:20:29,030 --> 01:20:29,040

 

2523
01:20:29,040 --> 01:20:29,830


2524
01:20:29,830 --> 01:20:31,350

 

2525
01:20:31,350 --> 01:20:33,830

 

2526
01:20:33,830 --> 01:20:35,910

 

2527
01:20:35,910 --> 01:20:39,189

 

2528
01:20:39,189 --> 01:20:40,629

 

2529
01:20:40,629 --> 01:20:43,430

 

2530
01:20:43,430 --> 01:20:46,390

 

2531
01:20:46,390 --> 01:20:48,470

 

2532
01:20:48,470 --> 01:20:50,709

 

2533
01:20:50,709 --> 01:20:50,719

 

2534
01:20:50,719 --> 01:20:52,709


2535
01:20:52,709 --> 01:20:52,719

 

2536
01:20:52,719 --> 01:20:53,990


2537
01:20:53,990 --> 01:20:58,470

 

2538
01:20:58,470 --> 01:20:58,480

 

2539
01:20:58,480 --> 01:20:59,270


2540
01:20:59,270 --> 01:21:01,830

 

2541
01:21:01,830 --> 01:21:03,669

 

2542
01:21:03,669 --> 01:21:05,510

 

2543
01:21:05,510 --> 01:21:07,030

 

2544
01:21:07,030 --> 01:21:10,229

 

2545
01:21:10,229 --> 01:21:10,239

 

2546
01:21:10,239 --> 01:21:11,430


2547
01:21:11,430 --> 01:21:15,350

 

2548
01:21:15,350 --> 01:21:17,270

 

2549
01:21:17,270 --> 01:21:19,830

 

2550
01:21:19,830 --> 01:21:22,470

 

2551
01:21:22,470 --> 01:21:24,070

 

2552
01:21:24,070 --> 01:21:25,669

 

2553
01:21:25,669 --> 01:21:27,430

 

2554
01:21:27,430 --> 01:21:30,470

 

2555
01:21:30,470 --> 01:21:32,870

 

2556
01:21:32,870 --> 01:21:35,430

 

2557
01:21:35,430 --> 01:21:37,189

 

2558
01:21:37,189 --> 01:21:38,870

 

2559
01:21:38,870 --> 01:21:41,350

 

2560
01:21:41,350 --> 01:21:43,189

 

2561
01:21:43,189 --> 01:21:43,199

 

2562
01:21:43,199 --> 01:21:56,229


2563
01:21:56,229 --> 01:21:56,239

 

2564
01:21:56,239 --> 01:21:58,320