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welcome back  i hope you had a good thanksgiving  and uh are all refreshed  and ready uh to  uh  uh think about some theory of  computation um we're in the home stretch  now this is uh we have this lecture and  two more to go and um  so today i have for you um  [Music]  a  completion of the  theorem we started before the break  um  where we had a uh  we where we introduced probabilistic  computation  and we uh talked about  the class bpp as i hope you remember  and we looked in particular at these um  these problems involving branching  programs  where we  started the proof that the  problem of equivalence of two read once  branching programs can be solved in this  class bpp so what i'm going to do is  spend the first  15 minutes or so just reviewing where we  were  because we started this um it feels like  a long time ago now  and i just want to make sure that you're  all uh  um  uh on the same page and we're all uh  um remembering what we were doing and  then i will finish off the proof  and  along  with doing that we're going to introduce  an important method um  well we started that we we looked at the  method of arithmetization last time so  we will review that  we're going to use that again in the uh  the  work that we're going to start on  thursday on interactive proof systems so  this is a kind of in some ways both a an  interesting theorem in its own right and  a warm up for what we're going to be  doing um on um  in the last topic  of the of the semester  um  okay um  so um  uh let's just  remember what we were doing so we  introduced probabilistic turing machines  um  so those are these machines that have  they're a kind of non-deterministic  machine but there's a different rule for  acceptance  and  these are also non-deterministic  machines which can either make  one choice  just to have a deterministic uh move at  a step or they can make two choices  um and when the machine makes two  choices we actually think of there being  a probability there um where the machine  is tossing a coin to decide which branch  to go on  um  so with that there is a  uh tree of possible branches  and the probability of some particular  branch is going to be one over two to  the number of coin tosses on that branch  um  and so  [Music]  we then use that to define the  probability that the machine accepts  um which is the sum over all of the  probabilities of the accepting branches  and the probability that it rejects is  one minus the probability that it  accepts  um  so you know thinking about it um  uh you know  this  captures the idea that if you just run  the machine  on a random  set of inputs  um  from the coin tosses  what the probability that you're going  to end up with the machine accepting  that's the probability of acceptance um  defined in that way  now if we're thinking about the machine  deciding some particular language  it's supposed to accept the strings in  the language and reject the strings  which are not in the language but  because of the probabilistic nature of  the machine it might get the wrong  answer on some branches  and so  we say that a machine decides a language  with a certain error probability means  that the probability of getting the  wrong answer is going to be at most that  error probability epsilon over all of  the possible inputs to the machine  so if we say that the machine is error  probability one-third that means that it  gets the right answer for every every  string with probability at least  two-thirds  um  [Music]  okay so  that led us to the definition of this  complexity class bpp  which i don't remember if i told you  what it stands for it's bounded  probabilistic polynomial time that's  what bpp stands for  and it is  the bounded means is bounded away from  one half  because we don't want to allow the  machine to have probability one-half um  because then um  bad things happen uh the machine can  just toss a coin when it decides to make  um  uh an answer and not really give us any  information  um  [Music]  then  we also uh went over  the amplification level we did not give  the proof but we went over the statement  of the theorem the proof is really just  a calculation  that you can  drive down that error probability to  something extremely tiny um  just by basically rep repeating the  machine and taking the majority vote of  what it does on several different runs  if you run the machine 100 times  and you see if it's mostly accepting  then you're going to accept and the  chances that the machine was really  uh biased toward rejecting even though  your in your samples see mostly  acceptance is extremely small and you  can calculate that but you can make that  very tiny so small that for all  practical purposes it's really giving  you the right answer um but you know  it's not deterministic so it's not quite  100  guaranteed  um  [Music]  and the way i like to think about um  bpp  in a uh in terms of the computation tree  of the machine um  so that when it's accepting  most of the branches are accepting  you know weighted by their probability  of course but most of the branches so  the there are many accepting branches  when you're accepting and many rejecting  branches when you're rejecting  so just another way of saying the same  thing  um now  we're gonna start right jump right in  with a check-in um and this is a little  bit more  not exactly the material of the course  but um a little bit more on the  philosophical side but let's just see  how you do with it um  when you're actually running  a probabilistic machine  you imagine the machine as we're as  we're kind of informally describing it  it's called tossing coins  um  every time it has a non-deterministic uh  every time it has a  a choice  so a choice tosses a coin to decide  which way to go of course  you know a real computer does not have a  coin to toss presumably well maybe you  might actually build  some hardware into the machine that lets  it access randomness in some sense you  know maybe it uses um some  quantum mechanical effect you know to  get some random value or maybe it uses  the timer i you know i'm not exactly  sure you know you can imagine having a  bunch of ways of implementing that  a typical way that people implement  randomness randomness  in an algorithm is to use a  pseudo-random number generator  which is a procedure that might give you  some kind of a value that looks random  but  may not actually be random  it's for example giving you  the digits of pi  if you express if you want binary  expressing pi as a binary  as a binary number  then you might calculate the different  successive digits of pi and use that as  for your random number generator um of  course that's a deterministic procedure  so it's not really random um  and but you know often people do use  those kinds of things when they're  simulating  random machines so what do you think  about doing that  could we use a random uh pseudorandom  generator  as the source of randomness for a  randomized algorithm um  what  yes or no or what do you think  um  so let's launch a poll on that  and see what your  opinion about using pseudorandom number  generators instead of truly random true  randomness for our algorithms  i'll give you a few seconds a  minute to  weigh in on that  okay  we're gonna close this down everybody's  participated who wants to uh one two  three  um  [Music]  okay  yeah i think probably the the best  answer is a  um let's take a look at some there are a  couple of answers here that  really that don't make um that aren't as  good i would say uh b  well you know usually people think of  random pseudorandom generators as pretty  fast procedures um they're not not that  interesting otherwise so  um i wouldn't say that b is a good  choice because  you know they're they're usually pretty  quick to implement  um  [Music]  c is  a worse choice even  because uh turing machines can do  anything that any other algorithm can do  so certainly if there is such a thing as  a pseudorandom number generator and  there is then you could uh implement it  on a touring machine um  d is kind of an interesting answer uh  because you're saying well that would  imply that p equals bpp if you could  actually simulate  uh randomness with a deterministic  procedure  um  but in fact the reason i i would  not choose d is because it's perfectly  conceivable that p  does equal bpp we don't know that p is  different from bpp so it's conceivable  that they're equal  and in fact i think if you if you polled  most complexity theorists  they would  most most people in my field would would  believe that p does equal bpp  just for this very reason  that if you had sufficiently good  pseudo-random number generators  you could actually eliminate the  probabilism in these probabilistic  computations you could just run them on  the pseudorandom number generator and in  fact there is some theory around that um  that has been developed um  but at the present time we do not know  how to prove that there are pseudorandom  gen number generators and it has some  actually there's actually in some line  of this research has some connection  with the p versus np problem but we  don't know how to prove that there are  sufficiently good pseudo-random number  generators that would would allow you to  run them on a probabilistic algorithm  and have a guaranteed behavior which is  as good as running truly random numbers  into the um  probabilistic  algorithm and so  the the answer that i would pick would  be a that you could use it sure you  might get the right answer but it's not  guaranteed we just don't know how to do  the analysis for the pseudorandom number  generators and you know if you had such  you had ones that were good enough they  would show p is equal to vpp  but that might be in fact the correct um  that might actually be true  okay  so let's continue on um  and uh  remember now branching programs you know  we had these uh kind of networks um of  nodes and edges  um and there was a procedure you know  we'll see a couple of examples again  some of the ones that we had from before  uh where you um  you know you have branching programs  look like this  and you know you you have a bunch of  query nodes you look at the settings of  the variables to decide whether to go  down a zero edge or a one edge and  eventually you're going to end up at an  output node and that's going to be the  output of the branching program and in  such a way these branching programs  define boolean functions  from the settings of the input variables  to a 001 output  now you might have two branching  programs and wonder whether they're  computing the same boolean function or  not and testing that is a co np complete  problem as you are asked to show on your  homework um  uh  now  if the branching program however has a  restriction namely that it's  not allowed to ask to query the same  variable more than once on a path  then  with that restriction you can  we call it a read once branching program  and then the situation for testing  equivalence seems to be different um in  fact we can give a bpp algorithm for  testing the equivalence of read once  branching programs even though such a  thing is unlikely to be the case for  general branching programs because of  the co-np completeness  okay so i hope you're comfortable and  with me on all that reasoning  um  okay  um  [Music]  let's uh  so  um  [Music]  all right so the idea for proving this  is what we're going to do is we're going  to take the two branching programs and  run them on a randomly selected input  but as we observed last time if you just  run them on a randomly randomly selected  boolean input where we assign the  variables zeros and ones  then that doesn't give you the right  answer with high probability  because the two branching programs  might be  different computing different boolean  functions but they differ only on a  single input setting  and then  just picking them at random you're not  going to have a very high probability of  finding that one place of difference  so instead what we're going to do  is define a way to run these branching  programs on non-boolean inputs where the  variables are set to values other than 0  and 1.  2 3 7 22  and make sense of that  and then argue that by running the two  branching programs on a randomly  selected non-boolean input  that that's very high probability of  giving you the right answer  um  so somehow by expanding the domain of  possibilities you're going to uh better  your chance of getting the right answer  very significantly  okay so even though these two branching  programs might agree on all of the  on almost all of the boolean  inputs  we're going to show that by  doing this arithmetization  so this is the method um  uh  they're going to if they're really not  equivalent they're going to differ  almost all of the time on the expanded  domain  okay now we have to prove  so that there is uh um  you know that's where today's work is  going to be  okay  so um  why don't we just stop  and  make sure we're all  together on this  i can take any questions um  i'll also review how the arithmetization  goes  but that i'll do that next  so are we all okay  on this  um  good so let's move on  so in order to move toward understanding  what it means to run the branching  programs on these non-boolean values  we're going to have to uh  uh get a somewhat different perspective  on the computation of a branching  program  so the standard com perspective is that  you know you um  you take your setting your assignment to  the input which is  0 1 1 for x1 x2 and x3  and use that to follow an execution path  through the machine  so you know x1 is 0 x2 is 1 x3 is one  the output is one  okay as i've indicated in yellow  um  this other perspective says well let's  let's we're gonna by operate by labeling  the nodes and edges of the machine  and  that's going to have a very direct  correspondence with the execution path  perspective so we're going to label all  the nodes and edges on the path with a 1  as indicated in yellow  and all the nodes and edges that are not  on the path all the other nodes and  edges are going to be labeled zero  okay  so by following the ones  it's like the breadcrumbs you know and  hansel and gretel these are the this is  the path you need to follow  to get uh through uh the machine  the zeros are the places where you don't  go  um  okay so the output label here  the the output is going to be the label  of the one  output node  whatever you're labeling that because if  it's a one that means the path went to  the one and if it's a zero that means  the path didn't go to the one who went  to the zero  okay so just by looking at this value  you can see what the output of the  machine is  all right so  um  let's  describe a different way of defining  this labeling without just looking at  the path  but it's going to capture exactly the  same thing  um  so we're going to say  whenever you have  you know if if you've already labeled a  node  i'll tell you how to label the two edges  that emanate from that node  um  [Music]  i'm going to label the one edge  a and  the query variable  why is that  well  a is going to be either a zero or a one  and it's going to tell us whether or not  the path  entered that node  so if it's a 1 it entered that node at 0  it didn't enter that node the only way  it's going to go down that this branch  here  is if it did enter  the x i node if it didn't enter the x i  know there's no way it can go down this  this branch so we're going to and that  value  so it's so the only way you can go down  this branch is if it went through that  node so that's the a the value of a  and  the x i is a one that's why we say a and  x i so you really have to understand  this little expression here in order you  don't understand that  you're toast  okay so you better understand this so we  can move forward i'm happy to take a  question these are the simplest  questions or sometimes the most valuable  if you understand why i'm labeling it  this way shoot me a chat  um  okay now the other brand uh other branch  i'm going to say well i'm only going to  go down this edge  if well a is true so i did go through  that node  and  x i is false so this is going to be a  and the complement of x i  all right so that's how i'm going to  label these this is another way  giving these expressions for labeling  these edges  based on the label of that node and  similarly in order to complete the  picture i've got to tell you how to  label the nodes based on the edges that  are coming into it  so if i know that i have a1 a2 and a3  which tell me  the status in terms of the path of  whether the path  went through any of these edges  well i know that it's going to go to  that node if it went it's the or of  these values  if the path went through here or it went  through there or went through there then  it's going to go through that note  that's why the or is the right thing to  say  so this gives me another way of  constructing the labeling over here  without even talking about paths  but i'm just as i describe it i argue  that it's going to get the same result  all right  so there's a question can we quickly say  again why we can't do that on boolean  i'm not sure i understand the question  so please send it to me again but this  is right now everything is boolean we  haven't done arithmetically anything yet  um  and the reason why we can't just live in  the boolean world is that  you know just by taking boolean values  of boolean assignments here we don't  have a high enough probability of  catching a difference between the two  machines  all right so let's continue um  all right so now i'm going to talk about  how to we're going to extend this  to the non-boolean case  okay using the arithmetic  arithmetization method so first of all  arithmetization is a simulation of and  and or with plus and times  such that if i think about  um  [Music]  true as a one and false as a zero  this is going to give me a faithful  simulation it's going to do the right  thing um it's going to compute exactly  the same values that we expect  so like a and b  well  um  times works just like and  um  and um right you know it does for  for one and zero  as true and false times exactly works  like and  um  and  negation is one minus  and or is going to be the sum  minus the product  and then these just give you the right  values  um a a or b  if you just calculate it out  by plugging in ones or zeros you get the  right answer just by using using this  arithmetic  um  so  now what we're going to do  instead of using the boolean labeling  we'll just use the arithmetical labeling  but it's going to compute exactly the  same thing because the arithmetic  simulates the boolean  um  so uh we always go through the start  nodes so there's no question about  labeling the very start node with a one  but now i'm going to give expressions  just like the boolean expressions but  now they're going to use plus and times  instead of ands and ors um  so let's just see remember what we  defined before we had a and  x i for this edge  i'm going to replace that what is and we  just look up here in our table in our in  our translation table  and becomes times  so we're going to replace that with a  times x i  and it's going to work exactly the same  but you know the difference is that  this makes sense  even when we have non-boolean values  times and plus are defined  for non-boolean values  um  where zans and or not  so what what goes down on this edge um  well this was  a and the complement of x i as you  remember  uh so we're going to that's going to  become a times 1 minus x i  and then similarly  we had or over here and here's a little  bit of a trick but that's going to be  important for the for the for the  analysis that we're going to do  instead of using the recipe for um or  in terms of plus and times we're going  to have something a little simpler it's  just going to be the song  and the reason why that works it's good  to understand  is that  because of the acyclic nature of the  branching programs at most one of these  edges can have a path through it  so this is a kind of very special ore  um  sometimes called a disjoint or  you're not allowed to have more than one  of the values be one because that never  happens in when you have an acyclical  graph you can never have the path coming  down this way and then again coming down  that way then it would be entering that  node twice had to be a cycle  so it's going to be good enough for us  and it necessary for us to represent  uh this or as as a sum  okay  so i think that's all i wanted to say on  this slide um so somebody's asking is a  possible sum of for some of these values  to be negative  yes the values that's a as it stands  right now some of these values can be  negative i haven't put any restriction  on what the values are are going to be  um so the input could be a negative  number and then you're going to just get  negative stuff happening uh in fact  there's negative there's subtractions  going on here so even with positive  numbers i think we did an example last  time i think we're gonna i'm gonna do  that example again uh of exclusive or um  where you're going to you get negative  numbers coming up that doesn't matter  but actually what we're going to end up  doing  is operate doing these calculations  modulo  some prime number q  okay um so i'm going to pick some prime  like  17 and do all the calculations mod 17.  um and the reason for doing it that way  is really because we're going to be  picking random assignments to the um  variables as our input  and it and you need to it makes it makes  the most sense to do that when you have  a finite set of possibilities to pick  them up  um so we're not going to pick like a  random integer there's infinitely many  possibilities and you could yeah you  could set up a distribution there but  that's very complicated that actually  might work i'm not sure i i haven't  actually gone through that analysis but  the typical way people do this is by  looking at what's called a finite field  so i'll talk about that in a second um  why is there at most one one among a1 a2  and a3 so um  the ones i'll say one once again but the  ones correspond to the path so this is a  one if the path went this way  just think about it the path cannot go  through a1 and  can at the same time go through a2  because that means the path went through  this node  then  how is it going to get over to a2 it's  going to go through that node twice  in an acyclic graph you cannot have a  path going through it's the same node  more than once  so you have you're going to have to  think about that  um  okay let's move on  so now we're going to talk about the  same um  uh  we're going to  we're going to look at that  non-boolean labeling applied to an  example so here is a  very simple branching program that  actually computes the exclusive or  function in the boolean world  um  so this is the labeling that we just  i just developed for you the the uh  arithmetical labeling um and we always  label the start node with one because  the path always goes through there um  and now let's look at this before we  jump ahead let's look at this edge here  remember what it is we have to apply  this rule here this is the one edge  coming out of a node that already is  labeled  so it's that that label on that node  times the x i because that if you're  thinking about that's the and captures  the end  so it's just a times x i so xor it's x1  in this case so it's going to be x1 is a  2  in our input  so it's going to be 1 which is the a  times the x one which is two so that's  just this this edge gets labeled two  now this well okay let's look at this  edge now i think that's next  this is the one minus x i it's one minus  x one so it's one minus two so we're  going to end up with a minus one one  minus times minus  one times one minus two which is minus  one so it's a minus one  now we're going to label these two nodes  using the other rule  um  so this gets a 2 because that's the sum  of the incoming edges this gets a minus  1 because that's the sum of its incoming  edges  and now we're going to look at um  uh  which order did i do this in okay it's  going to do this uh  this edge now  um zero edge which is going to be 2  times 1 minus  its variable so it's 1 minus x2 x2 is a  3  so it's 1 minus 3  it's a minus 2 so 2 times minus 2 is  minus 4.  okay i don't want to rush through  there's no point in just blabbering on  i'm just trying to do work this example  so you get the idea  okay so could you fill this out this  next one out yourself  so this is the one outgoing edge from  this x2 node  um so x2 is a x2 is labeled  uh  with two here so there's a is two  this is the one edge going out so it's  the one on this side here so it's  two times the  x2 x2 is a three so it's two times three  so this should be six  oops right here six  i'm gonna do the same thing so  x2 is labeled minus one  um  so minus one times one minus so you got  a two here  you get a minus three here just  following again the same um process  and now let's let's say what is the  label on the zero node here so you again  you take the sum of its incoming labels  so there was a two  and a six  so that's an eight  and this is a  uh  a minus three and a minus four coming in  so it's a minus seven what's the output  the output is minus seven because that's  the label on the one node  okay output is minus seven  okay  so this is going to be our  way of  defining  the  output  of a branching program when it has a  non-boolean setting on its inputs  if you had the boolean setting on its  inputs  and you followed this procedure what  would you get out  you would get the same answer that you  would get by following the path  because the  arithmetical simulation is a faithful  simulation of the ands and ors and the  ends and auras capture exactly when the  what the path does  so this is a strict extension  of the operation of the branching  program into a new realm these  non-boolean values on the old realm it  behaves just as it would or  did originally and that's critical  to understand that  um  yeah somebody's asking would the final  value of the zero state also be the same  sure  in the ba in the in the boolean case  if we follow this in the boolean cases  all of the labels would be exactly what  they were  in the bool you know  you know they would just be the zeros  and ones that we had from before  so does this exactly mimic xor if you  take this all mod  2  hmm  i don't know i'd have to think about  that i don't think that that's essential  um in this case it might behave  correctly if you take the answer mod 2  for xor but the xor is going to be very  special i'm not sure that might happen  to be true i'd have to think about it  for a second but um  i'm not sure that's in  relevant  um  okay this is a good question if i picked  a different branching program that also  implements xor  um so it would be an equivalent  branching program  would it behave the same way  on the non-boolean values  and the answer to that is  yes and no  you understand the question it's a very  good question and it's actually we're  going to prove this  you know  in the analysis  it's going to be easy to prove because i  gave you both possibilities yes and no  but let me tell you what i mean by that  so um so you understand the question  suppose i had a different branching  program i'm not sure you can come up  with a different branching program but  let's make let's let's let's say you  could you have a different branching  program yeah sure you can come you can  do the variables in a different order  for example so you suppose you come up  with a different branching program that  gives you  xor  and now you plugged in  two and three  would i always get the same value out  yes  if that other branching program was read  once  no  not necessarily if it's not read once  and that's why read once is critical  as we will prove  for two read once branching programs if  they behave the same way  on the boolean values  they behave the same way  even on the non-boolean values  that's not necessarily true if the  branching programs are not read once  okay so let's we will see that we will  prove that  and use that that's going to be  important  okay so here is the algorithm sketch  which is kind of a little bit even sort  of suggested by that uh that that  that very good question um  uh  so what we're going to do is we want to  take the um the two branching programs  for with that we're trying to test if  they're equivalent  we're going to pick a random non-boolean  input assignment  so set the values here  chosen from the field but we'll get  there  you know  a random value for x1 a random value for  x2 and so on these could be numbers like  17 and 25 and 23 and so on  and then run  using this process run the branching  programs  to evaluate them on that non-boolean  assignment  if they  disagree then we're going to know that  the two branching programs were not  equivalent even on the non-boolean case  even on the bullying case did i say that  wrong so if they disagree even on the  non-bullying case they have to be an  equivalent even in the boolean case  not obvious  but if they agree  and if they agree  then it's not a guarantee that they're  equivalent but it's going to be very  strong evidence that they're that  they're equivalent  okay so that's where the probabilistic  nature is going to come in  okay so we're going to prove that  so first we have to develop an algebraic  fact  um  and that involves polynomials  this is this is a simple thing that i  think many of you have run into already  perhaps even you know in high school um  uh i'm not going to prove the algebraic  facts but i'm going to state them and  actually the proofs are not hard they're  in the textbook um  okay so let's suppose we have a  polynomial  of degree d here it happens to have b so  you know some kind there's a bunch of  constants these a's are constants  x is the variable of the polynomial  and uni i presume you've seen  polynomials written out like this  and so if you have if you assign x to  some value some constant value z  um  and the polynomial value evaluates to  zero  we often call that a root of the  polynomial  um so the polynomial these are the  places where the polynomial evaluates to  zero and i've kind of shown over here  those are the roots  it's not hard to show  that a low degree polynomial cannot have  lots of roots  basically if the polynomial has degree  at most d it can have at most d roots as  long as the polynomial itself is not the  everywhere zero  polynomial because obviously then  everything is a root  oops typo thank you should be d minus  two over there  uh  pretend that's a two  uh  all right so um  all right so if we have a low degree  polynomial let's not get ahead of  ourselves below degree polynomial uh  degree did you do the polynomial of  degree of most d it has the most d roots  and that's a simple proof um the basic  idea is every time you have a if you  have a root of the polynomial you know  so if um  uh  if setting x equal to five  gives you a root of the polynomial um  you know it's a zero of the polynomial  then x you can easily see that x minus  five is a factor of the polynomial you  can divide out x minus you can divide  out  by x you can divide by x minus five and  you get a new polynomial of degree one  less  and you can just uh which is going by by  induction has a one fewer uh root um so  you can just divide out by the roots  basically it's very straightforward  uh  um  uh  okay so now um other very important  thing  uh  if i have two polynomials  that are both low degree they cannot  agree  on very many places  that follows right  from the from what i just proved above  because what i'll do is i take those two  polynomials and i look at the difference  which is also a low degree polynomial  every time there's an agreement between  those two original polynomials there's a  zero of the difference polynomial  and because that difference polynomial  cannot have too many zeros the two  original polynomials cannot have too  many agreements  okay  so the corollary is that if x and y are  both polynomials of degree of most d  and they're not the same polynomial  because then the difference would be the  zero polynomial  then the number of places where they  equal is the most d  okay so the proof is just letting p be  the difference of p one and p two  very standard kind of a trick  okay now the the above is going to hold  for any field so we're gonna be a field  is just a set with plus and times with  the familiar  um properties  the distributive law and so on and  identities and all the stuff that you  would expect plus some time to have in  in the normal world  um and so we're going to talk about  finite field  that has only q elements  um  that has exactly two elements where q is  a prime number so it turns out and i'm  not going to prove all this but it's  pretty simple stuff  that if you just take the numbers from  one to q minus one  from zero to q minus one um they uh and  use plus and times mod q  that that does has all the right  properties to be a field  okay so we just think of it it's just  modular arithmetic  mod some prime q  okay and then it's we can in a natural  way pick  a random  assignment  to  a variable from the field because it's  just choosing from among queue  possibilities  so uh  uh  yeah so the  yeah so getting a question here the the  the the coefficients of the polynomial  and the assignment to the variables  they're all going to come from this  field  so everything's going to be we're all  going everything is going to be  operating uh  in this field don't get don't let that  throw you off or you're just your  ordinary intuition about the way  arithmetic works is going to be just  fine  um  okay  now  if we have  ah but this is important here from the  perspective of thinking about this  probabilistically so i'm going to  rethink about this polynomial lemma  which says there are not too many  roots  um in terms of the probability of  picking an element of the field what are  the changes that it happens to be a root  so if you have a low degree polynomial  and you pick a random value in the field  what's the probability that you got a  root  well it's just the number of roots  divided by the size of the field  so if you have this really big field and  you have this low degree polynomial it's  going to be pretty unlikely that that  you're going to end up picking one of  the zeros  one of the roots  uh just at random  that's all that's this is saying  so there's the most d roots out of the q  possibilities  and the last thing i'm going to  introduce here is the multivariable  version of this  um which is called  perhaps somewhat unfairly but uh it's  called the schwarz zipple lemma  though in various forms it had been um  known  prior to their work um  in some cases  you know the literature actually goes  back  a long ways but anyway the  this is called the short zipline it  doesn't really matter um uh  except to the people whose  um  uh  credit is being denied but  that's not one of us uh so anyway the  schwarzepilema  says that if you have now a polynomial  in several variables which is not the  zero polynomial  where each variable has low degree so if  i say if has degree at most d in each x  i  so each variable is going to have at  most an exponent of d appearing in that  polynomial  and now if we pick  random values  to assign to all of those m variables  from the field  the proper probability that we ended up  with a root  and we ended up with a zero  is something you can bound  so it's m times d so the number of  variables times this maximum degree  divided by the size of the field  and this is going to come up later for  us  and this is another  fairly simple proof  a little bit more sophisticated than the  one that we had above and in fact she  uses that one as a um  as a lemma to prove this theorem  okay so we're going to uh not going to  prove any of that but i refer you to the  book if you're curious  um  uh  so  um  yeah so like a couple of good questions  here  what happens if these values are bigger  than q for example then then then it's  not telling you anything if d is bigger  than q m is bigger than q or the product  is bigger than q  then  then you learn nothing  uh from this lemma from this theorem  so  typically what the in applications  you're going to pick a larger you're  going to have the flexibility you get to  choose q to be in something that you  want so we're going to pick the field to  be big enough  so that the m and the so the m and d are  going to be relatively small  um in fact d is going to end up being  one as we will see  um and m is the number of variables so  we're going to pick q which is going to  be substantially larger than the number  of variables  and how is the degree defined in  multi-variable polynomials you know the  m if the polynomial has  you know  x y squared plus  3 x to the fifth y squared z you just  you just pull out each variable  separately  and you look at the maximum degree of  that variable so the x in that case had  had a degree five  appearance the y had a degree two  appearance so you take the maximum over  all of the variables of the degree of  that variable and that's going to be the  bound on the degree of the polynomial  polynomial  so in fact in our case d is going to be  1.  so all of the variable there's not going  to be any exponents on anything  everything's going to be exponent one  um  so q  is is q related to the number we choose  for the mod yeah q is the number we're  choosing for the mod  we're doing everything mod q  so all of the arithmet all the  arithmetic is going to operate mod q  and that's the size of the field  that we're going to pick  okay so i think we're here at the break  um happy to take some more questions um  but why don't we just start that off  uh  and uh  i will see you in five minutes but in  the meantime happy to shoot me questions  okay so what happens if we use boolean  assignments in the xor example  um  so the x you know would that work  um to be able to check  uh uh agreement  um  it would it was so it's hard to it's  hard to  make an argument based on just a single  example  i think the better thing would be  to look at  you know two um  um  branching programs that just differ in a  single place so i can even suggest to  you can make a branching program that  always outputs um  true  it doesn't even read its variables or if  it reads them and they always go to the  same place and then ends up always at  the one output so imagine a branching  program that always outputs one no  matter what the assignments to the  variables are and you can you can easily  make such a thing  and then you make another branching  program that computes the or function  so it reads every variable  and  it's going to be  1 if any one of those variables is set  to 1.  okay so  the only time the or function is 0 is if  everything is set to zero  but  now if you're trying to randomly check  whether the always one function  is equal to the or function  of course without knowing in advance  what they are because that's cheating  you're just given these two functions  and you want to know these two branching  programs  and you want to know are they computing  the same thing or not  and by this procedure of randomly  sampling  there you're always going to get these  branching programs both to say one  unless you just happen to pick the  random assignment of everything set to  zero  and that's very unlikely that you're  gonna pick that you know if you imagine  you have your branching program has a  hundred variables in it it's only two to  the minus 100 chance that you're going  to set them all to zero  randomly  and so you're almost you go you're  extremely unlikely to find the one place  of difference if there's only a single  place if there's lots of places of  difference then you  you know it's not not so bad but if the  number the fraction of differences is  low  you're going to have to do a lot of  samples in order to find that it won't  be in you know possibly exponentially  many samples and then you won't run a  polynomial time okay so let me just see  if there's other questions here um  why can we accept  going back to the boolean labeling side  why can we accept that b1 equals b2 if  b1 and b2 agree on  only on just one input assignment no we  didn't say that  um all right i'll go back there boolean  assignment  is this i'm not sure which one you mean  is this one what do you mean  i don't know  boolean labeling site must be it  why do we  non-boolean labeling okay  so what  no  i see what you're saying you're saying  about this here we're just going to pick  one random assignment  and if they agree on that one random  case  then we will say um accept  because you might think well we should  take to a whole bunch of samples that's  a good question  um but in fact we're going to arrange  the probability such that if the two  things are not equivalent  then you're  it's going to be uh diff the value the  values will be different almost  everywhere a large number of places so  just picking one and having them agree  is going to be strong enough evidence  that you're still going to accept and  you'll have a still a low probability of  getting an error  you'd have to see the analysis  um  are we assuming the roots of the  polynomials are integers no  don't forget when operating over a field  here so it's not even talking about  integers doesn't totally make sense but  it doesn't really matter um we're not  assuming that the bound still  uh  the bounds still oh i should have taken  this away um  the bound still works  um uh  the bound still works  even even if we have non-integers  okay i'm not sure if i'm being helpful  here why don't we just move on  um but we're not assuming that these are  integers because the bound doesn't  doesn't matter if  it says there's at most five roots  including the reals there's still going  to be five roots including the integers  all right so let's continue  um  [Music]  good  all right  so now  everybody's back i hope  let's talk about  moving forward here  um we want to basically what where we're  going with this is we want to  analyze  the algorithm  which picks a random non-boolean input  and evaluates the two branching programs  and in order to do that  we're going to look a little bit more  carefully  at the what happens  uh when we  arithmetize the branching program  and um we run it on these non-boolean  values and so what i'm going to do  is  take these take this branching program  let's say this is the same  xor exclusive or branching program but  instead of  labeling it  as we did before by setting x1 to 2 and  x2 to 3  i'm going to leave x1 and x2 variables  and just do a symbolic execution  so i'm going to label these things  just leaving x1 and x2 as variables  okay so let's just see what we get if we  do that  so  remember we assigned this to be one  now  this edge here  is  you know here's the rule it's a which is  1  times  x1 so this should be  without knowing what the value of x1 is  leaving it as a variable we're just  going to put down x1 over here  over here what goes over there  well it's 1 times 1 minus x1  just 1 minus x1  now we're going to add things up as we  did before  and now what happens for example i think  i have this edge coming next let's look  at this edge the one edge coming out  so that's the label now is x1 on this  node  and the out this is the edge this is the  one edge coming out so you multiply by  the um by the value of x2 that we're  leaving it as a variable so you're just  going to multiply it by x2 and so we're  just going to get x1 times x2  x1 x2  on this edge now what happens on this  edge  so this x1 times think with me  um  times  1 minus x2  and similarly over here we have 1 minus  x1 on this node  so i think on this  on the one edge coming out it's 1 minus  x1 now times x2  because that's this rule again  1 minus x1 times x2 and this is going to  be 1 minus x1 times 1 minus x2  now we're going to add this up  for the zero  node  so we have this this value 1 minus x1 1  minus x2 plus x1 x2  and on this node here we're going to um  add these two values up  so 1 minus x1 times x2 plus x1 times 1  minus x2 and that's the output now  expressed symbolically  now you could plug things in and you're  going to get the same value out as you  did before  um  but let's leave it as a polynomial  for now uh because that's going to um  [Music]  help us and analyze this  so now notice the form of this  polynomial is something special  um  what happens is it's gonna look like a  bunch of products  of  x i's and one minus x i's  added up so it's a sum of products of  that form  so each row here is a product  of x i's and one minus x i's  and then those rows are all added  together  i claim that's going to be the form  of this polynomial you see this already  has that form  um  and the reason for that is  every time  you go to a node you're just adding  things up so that's just going to be  like adding adding up more rows  and every time you go down to through an  edge you're multiplying what you have so  far either by  an x i or a one minus x i  so you're just accumulating products of  x i's and one minus x i's and you're  just adding them up so this is what that  polynomial is going to look like  okay  now let's look a little bit more  carefully  at  the form of this  so for one thing  could we have  um  higher powers here  like a x2 cubed  could that happen  so because you know when i say it's  products of x i's and one minus x i's  maybe there's going to be some x xi's  that appear several times in the product  well that cannot happen  why  read one sprint it's a read once  branching program  so every time you multiply by an x i  or one minus x i  you're never going to do that again  because  doing that would imply you're querying  that variable more than once  so this can't happen  okay so i crossed that and this is sort  of appeared off to the side here but  yeah i'm crossing that out that does not  happen  another thing  that  is part of that's worthy  that's going to be helpful to to notice  about this  this polynomial and by the way maybe i'm  i'm being confusing here this is i'm  supposed to be  representing sort of as a generic form  of the polynomial this is not some  particularly i shouldn't have said this  at the beginning this is not some  particular polynomial that came from  anything i'm just trying to describe  what the general form of the polynomial  looks like  just as an illustration  so you know it's so this polynomial is 1  minus x1 times x2 times 1 minus x3 times  x4  and so on  and and adding up a bunch of rows like  this i'm just saying this is what the  polynomial will look like  for maybe some branching program so  every branching programs that are going  to have some polynomial that looks sort  of like this  um  and what i'm also going to say  what for convenience now i want to say  that each row is going to have every  single variable up here  either as an  x i or as a 1 minus x i  so in order to get that i need to make a  further minor assumption about the  branching program that it's a read  exactly once  currently when i say read once it can it  can avoid reading some variables on some  branches because i could read it most  once but now i want to say that every  variable gets read exactly one time on  every branch  and what that's going to mean is that  every row is going to contain every  variable  either  as an x i or as a 1 minus x i  we can eliminate that extra assumption  easily and i'm going to leave that as an  exercise to you it's not very hard to do  um so i think if you follow me you can  see  and you play with it for a minute or two  you'll see that it doesn't really matter  uh you can either you  but i think just for the for the first  time through this  let's assume that every row has every  variable  okay so important to understand um  okay so  so the this is the output polynomial of  this branching program so  let's look further more at this  polynomial  and understand  the rows  let's take one one row  out of this polynomial  and to understand its um  uh  um  understand what it represents  so one row here it's a product of a  bunch of things  part of a bunch of variables either  variables or one minus the variables  now  notice  let's think about this in the boolean  setting first of all  so in the boolean setting  each of these variables are going to be  zeros and ones  and the one minus the variables are also  going to be zeros and ones so it's going  to be a product of zeros and ones  if there's a zero that appears in that  product  that product is going to be a zero  because zero times anything is a zero  so the only way that product cannot be  um zero  is if all of those  uh  values are once in the boolean case  so  uh that means that  x1  well let's  let's let let's look at the second row  so x1 had to be a one x2 had to be a one  x3 had to be a one x4 had to be a zero  um in order to continue the product of  ones  and so  on  so in fact there's only a single boolean  assignment  to these variables which make that row  one  every other assignment to those  variables makes that row zero  saying that another way  these row each of these rows corresponds  to one of the rows  of the  truth table for the boolean function  where  where the truth table is true gives a  true value gives a one value for the  function on that row  okay so i i hope you've all familiar  with the notion of a truth table of a  boolean function you just write down the  boolean function  every possible assignment to the boolean  function and you write down  one or zero or true or false for what  the value of that function is so just a  tabular representation of the boolean  function it's called the truth table  this thing here gives you all of the  ones all of the rows that are one in  that boolean function that's what this  polynomial gives you  okay so i think we're  at a pause point  for  um this slide and i  um  it's deathly silent on the chat  so i was feeling that that was a  that went down that went down rough  uh for for you um it's it's we're gonna  it's important to understand the form of  this polynomial  uh  here um  you know that it's corresponds to the  truth table of the boolean function so  each one of these rows is only going to  be on again thinking boolean now  each one of these rows is only going to  be one  on that on an assignment which makes the  function one  um  one of the one of the assignments that  makes the function one  okay and somebody says and similarly for  the expression for the zero node yeah  the zero node which i'm not focusing on  but yeah the zero node would be all of  the false rows of the truth table  but the one node the polynomial for the  one node are all of the true rows  correspond to all of the true rows in  the in the in the um  in the um  uh  in the  in the  function of the branching the function  that branching program computes now  the reason why this is let me just tell  you where we're going  is it possible to have two rows that are  the same  if you think about how the rows are  being produced  no you can't have two rows of that are  going to be the same um  uh  because for one thing  you have to think about what what this  looks like when the boolean case if you  have two rows of the same that means  this thing is going to have an output  which is non-boolean  um  because you're going to end up with a 2  coming out that way  by adding those rows together  they can never happen and if you just  look at the way it's constructed you're  never going to because every time you  have a branching  you know one way is a an x one an x i  the other one is one minus x side so  every time you're branching there every  not every time there's a node they're  different um so you're never going to  have two rows that are going to be the  same but  let me tell you the importance of  connecting up  the this polynomial with the truth table  um because that tells us  that if the two functions of the two  branching programs that we started off  with  agree in their boolean values  then the two polynomials are going to be  the same  um  [Music]  you know because if the two branching  programs  uh have the same  boolean function so they're equivalent  then the um  truth tables will be the same and  therefore these polynomials will be the  same  so that and therefore they will behave  the same way on all  non-boolean values because they're the  same polynomial  so i'm sort of getting ahead of myself  but that's where that's what we're going  to argue that's why it's important to  understand the connection with the truth  table  because it sort of builds on the  um  under you know understanding something  about how this thing behaves in the  boolean case is going to give us  information about how it behaves in the  non-boolean case  okay  let's continue here then  um  okay so this is i think is this my  um  yeah this is essentially the last slide  okay um  we're going to spend some time on this  one so here's the algorithm  we are going to take our two branching  programs  the variables are x1 to xm  first of all we're going to find a prime  which is at least  three times  m the number of variables  m is not a very big number  um it's just a number of variables so  finding a prime that's bigger than that  is straightforward we're not talking  about huge primes here we're talking  very modest size primes so you can just  you know just try them  even trial and error is going to be good  enough um  now  that's going to be the  the size of the field it's going to be  field of size q um  and now we're going to pick  a non-boolean assignment to the  variables  and  okay we're going to pick a non-boolean  assignment to the variables  and  [Music]  evaluate the two branching programs on  that non-boolean assignment using the  arithmetization  if they agree  then we'll accept if they don't agree  then we're going to reject  okay now we have to argue that this  works  so we're going to first of all  arithmetize these two branching programs  i'm going to get the two polynomials  they each have the form that as i  described  um so a bunch of rows that correspond to  the truth tables of those two respective  branching programs  okay first claim  that if the branching programs were  equivalent so they compute the same  boolean function  then  the two polynomials agree everywhere  um  so then the two branching programs are  going to get the same value  um  on every non-boolean case as well as on  the boolean cases so they agree on the  boolean that means they always agree  even on the non-bully and i kind of  argue that already  but okay the other point is that if the  two branching programs  are not equivalent so they  differ  at some boolean value  now picking a random value to eval for  the polynomial evaluation you're going  to have only a one-third chance that  they're going to agree  so a small chance  all right so now um  let's prove these two facts the first  one i already kind of argued if the two  branching programs agree on all the  boolean values  um  then their functions have the same truth  table  so then the polynomials are identical  because the polynomials correspond to  the truth table  and so therefore they always agree  even on the non-boolean values  okay so  that means that the probability that if  you evaluate the two polynomials on a  random place  the it's good  whether you'll be equal that's a  certainty  because in fact in this case p1 and p2  are the same polynomial  now for two  if the  branching programs differ somewhere even  in one place  well you know the polynomials could not  be the same  they have to be different polynomials  because the polynomials include the  behavior of the  in the boolean case as well as all the  breadth of the field  um  so the polynomials have to be the same  and now we're going to apply the short  zipple lemma  we have two  different polynomials  they can only agree in a relatively  small number of places  so that says that  from the short simple theorem then um  the probability that p1 and p2 agree at  this random location is at most this  value that we had from before  the degree times the number of variables  divided by the size of the field  the degree is one  and the field is at least three times  uh the number of variables in size so  that means um  you get this inequality here and so  therefore the probability is it is a  third  and most a third  and that's good enough  um this is the probability that you get  the wrong answer it's going to be at  most one-third  so you're going to get the right answer  with at least two-thirds probability so  even just doing a single sample point is  going to be enough to to give a bpp  algorithm  all right so that's  um  what i have is are a couple of check-ins  here  uh now for you  um  [Music]  whoops  somehow this  better take me out of here um  uh  all right  so this is this is a little hard but  let's see how you do on it  suppose the branching program is  uh  well maybe i sort of a little bit  discussed this already that's okay the  branching problems will not read once  the polynomials might have exponents  bigger than one so where would the proof  fail  um would they fail the point where  b1 and b equivalent to b2 implies that  they agree on all boolean inputs so  that's sort of this  um the first step here  or  um  is it that agreeing on all boolean  inputs implies that the polynomials are  the same  or would it be that having the two  polynomials being equal implies that  they always agree  so those are the three sort of steps in  the  part one of the the proof of part one  so let's see uh what do you think there  in a couple questions about picking the  prime number  um  the this uh the prime number here is uh  it's this is a very small prime number  um  you could even represent that prime  prime number in unary  within the amount of time we have  because don't forget this is a prime  number whose magnitude  it is at most the number of variables so  you can write that prime number in unary  and um  uh  finding the prime and testing primality  of testing whether the number is prime  is something that you can do even with  this brute force algorithm  and it would be good enough  um so there's not there's no you don't  have to do anything fancy about testing  primality um in this case  okay so why does it have to be prime  you need it to be prime in order for the  for the um  for it to be a field  so this is just the algebra part  um  if you did not have a prime number then  then some of the field properties don't  work and you may no longer get the um  the fact that the polynomial has a small  number of roots  um  so that  that's all i can say about that is the  polynomial like a hash function for the  branching program are the equal if they  are the same but sometimes the values  also equal if the programs are different  that's an interesting idea so is that  is a polynomial acting like a hash  function i think it's there is something  to what you're saying but i think it's  actually in the other direction um  uh  it's it's actually more it's related to  a hash function but it's actually acting  more like an error correcting code  let's let's save that for later it's a  very good point it's a very good  question maybe we can talk about it  after if you remind me  um  okay let's let's end this poll here  um  should c be  if  having these two agree implies well i  mean  the question is should should we change  p1 and p2 always agree to b1 and b2  always agree well b1 and b2 are behaving  exactly the way b p1 and p2 behave so  i'm not sure it matter really matters  um  uh  too much time on this chat on this poll  here let's end this poll  oh sharing results  um  yes so the correct answer is b  um  that agreeing on all boolean inputs  implies that they're equal  um  the other two are follow kind of  immediately they're still true  but  if it's not read once even though they  agree and all the boolean inputs they  won't necessarily agree  as polynomials  um for one thing like if you just take  the two polynomials  x1 squared and x1  they agree on all the boolean inputs but  they're not the same um  they agree in the boolean world because  zero squared is zero and one squared is  one but they they're not the same  polynomial  okay uh let's move on actually i have  another  uh check in on the same slide here  um  and this is actually answering a  question that we i got in the chat if  p102 where x  you know  how big how big are these polynomials  are they these  look like they could be big if they're  exponentially large would that be a  problem for the time complexity  so  pick a or b  here we're running out of time here so  why don't i uh  not say too much and just let you go  with it  okay  i'm gonna close this down  all in  well  my god  by one point  this is a  this is  uh  this is like georgia here um but do a  recount um  in fact  b was is correct  uh  by it by by a hair here um they are not  polynomial in size the truth tables can  be very large but we don't have to  as we did with the um  branching program for  um  the exclusive or case you don't actually  write down the polynomials to evaluate  them you can evaluate them as you're  going along the polynomials are huge  but you don't have to write down the  polynomials to evaluate them that's only  part of the proof  you don't have to re either the the  algorithm doesn't have to deal with the  polynomials itself  so  maybe a  good uh  good to think about this  that somebody says did i invent this  proof no i did not invent this proof  it's a wonderful proof but it's not mine  i would love to have been get take the  credit for it  okay so um  why don't we uh  wrap this up  so you know can we just is there some  way to simplify the polynomials as we're  going along so that we we don't end up  with them being too big um  and so that we can then just look at the  polynomials not that i know of  um  uh i i i think the polynomials are  really going to be big  and so there's not going to be any way  to um  uh to to view it just as pawn if you  could it would be fantastic because that  would give you a deterministic algorithm  um  i think the only way that people know  how to do this is terms of you know  random inputs the polynomial to a  polynomial which is too big to write  down if you could write it down and just  analyze the polynomials  you'd have a huge huge result there  um  okay  i'm glad  people like this proof that's good  um  how many actual quantities are there in  the formula i'm not sure what that means  what formula  um i mean the polynomial is huge  uh the number of different polynomials  well  i i guess i don't understand the  question what motivates the idea of  arithmetization  what would make somebody think of this  i'm not sure  actually  um  but we're going to use it even in a more  remarkable way  uh in the in the last two lectures of  the course  so stay tuned  i mean this is sort of a clever but  seems very specialized but the um  the next part  where we're going to next application  we're going to use this to analyze  satisfiability which is much more  general kind of a situation and that's  that that i think is especially  remarkable  um  um  for the polynomials  p1 and p2  uh it's only a polynomial number quantum  i don't think so because well it depends  on the size of the field the size of the  no it's going to be something like m to  the n m power right  um so those are the number of possible  inputs  so it's in each each each field element  is  is  has  3m possibilities roughly and there are m  field elements so it's m to the 3m  different possible  inputs  that you're picking at random  so anyway i'm going to shut this down  and move over to the  to the others the office hours zoom  feel free to join me there otherwise i  will see you all on thursday  take care  you

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welcome back  i hope you had a good thanksgiving  and uh are all refreshed  and ready uh to  uh  uh think about some theory of  computation um we're in the home stretch  now this is uh we have this lecture and  two more to go and um  so today i have for you um  [Music]  a  completion of the  theorem we started before the break  um  where we had a uh  we where we introduced probabilistic  computation  and we uh talked about  the class bpp as i hope you remember  and we looked in particular at these um  these problems involving branching  programs  where we  started the proof that the  problem of equivalence of two read once  branching programs can be solved in this  class bpp so what i'm going to do is  spend the first  15 minutes or so just reviewing where we  were  because we started this um it feels like  a long time ago now  and i just want to make sure that you're  all uh  um  uh on the same page and we're all uh  um remembering what we were doing and  then i will finish off the proof  and  along  with doing that we're going to introduce  an important method um  well we started that we we looked at the  method of arithmetization last time so  we will review that  we're going to use that again in the uh  the  work that we're going to start on  thursday on interactive proof systems so  this is a kind of in some ways both a an  interesting theorem in its own right and  a warm up for what we're going to be  doing um on um  in the last topic  of the of the semester  um  okay um  so um  uh let's just  remember what we were doing so we  introduced probabilistic turing machines  um  so those are these machines that have  they're a kind of non-deterministic  machine but there's a different rule for  acceptance  and  these are also non-deterministic  machines which can either make  one choice  just to have a deterministic uh move at  a step or they can make two choices  um and when the machine makes two  choices we actually think of there being  a probability there um where the machine  is tossing a coin to decide which branch  to go on  um  so with that there is a  uh tree of possible branches  and the probability of some particular  branch is going to be one over two to  the number of coin tosses on that branch  um  and so  [Music]  we then use that to define the  probability that the machine accepts  um which is the sum over all of the  probabilities of the accepting branches  and the probability that it rejects is  one minus the probability that it  accepts  um  so you know thinking about it um  uh you know  this  captures the idea that if you just run  the machine  on a random  set of inputs  um  from the coin tosses  what the probability that you're going  to end up with the machine accepting  that's the probability of acceptance um  defined in that way  now if we're thinking about the machine  deciding some particular language  it's supposed to accept the strings in  the language and reject the strings  which are not in the language but  because of the probabilistic nature of  the machine it might get the wrong  answer on some branches  and so  we say that a machine decides a language  with a certain error probability means  that the probability of getting the  wrong answer is going to be at most that  error probability epsilon over all of  the possible inputs to the machine  so if we say that the machine is error  probability one-third that means that it  gets the right answer for every every  string with probability at least  two-thirds  um  [Music]  okay so  that led us to the definition of this  complexity class bpp  which i don't remember if i told you  what it stands for it's bounded  probabilistic polynomial time that's  what bpp stands for  and it is  the bounded means is bounded away from  one half  because we don't want to allow the  machine to have probability one-half um  because then um  bad things happen uh the machine can  just toss a coin when it decides to make  um  uh an answer and not really give us any  information  um  [Music]  then  we also uh went over  the amplification level we did not give  the proof but we went over the statement  of the theorem the proof is really just  a calculation  that you can  drive down that error probability to  something extremely tiny um  just by basically rep repeating the  machine and taking the majority vote of  what it does on several different runs  if you run the machine 100 times  and you see if it's mostly accepting  then you're going to accept and the  chances that the machine was really  uh biased toward rejecting even though  your in your samples see mostly  acceptance is extremely small and you  can calculate that but you can make that  very tiny so small that for all  practical purposes it's really giving  you the right answer um but you know  it's not deterministic so it's not quite  100  guaranteed  um  [Music]  and the way i like to think about um  bpp  in a uh in terms of the computation tree  of the machine um  so that when it's accepting  most of the branches are accepting  you know weighted by their probability  of course but most of the branches so  the there are many accepting branches  when you're accepting and many rejecting  branches when you're rejecting  so just another way of saying the same  thing  um now  we're gonna start right jump right in  with a check-in um and this is a little  bit more  not exactly the material of the course  but um a little bit more on the  philosophical side but let's just see  how you do with it um  when you're actually running  a probabilistic machine  you imagine the machine as we're as  we're kind of informally describing it  it's called tossing coins  um  every time it has a non-deterministic uh  every time it has a  a choice  so a choice tosses a coin to decide  which way to go of course  you know a real computer does not have a  coin to toss presumably well maybe you  might actually build  some hardware into the machine that lets  it access randomness in some sense you  know maybe it uses um some  quantum mechanical effect you know to  get some random value or maybe it uses  the timer i you know i'm not exactly  sure you know you can imagine having a  bunch of ways of implementing that  a typical way that people implement  randomness randomness  in an algorithm is to use a  pseudo-random number generator  which is a procedure that might give you  some kind of a value that looks random  but  may not actually be random  it's for example giving you  the digits of pi  if you express if you want binary  expressing pi as a binary  as a binary number  then you might calculate the different  successive digits of pi and use that as  for your random number generator um of  course that's a deterministic procedure  so it's not really random um  and but you know often people do use  those kinds of things when they're  simulating  random machines so what do you think  about doing that  could we use a random uh pseudorandom  generator  as the source of randomness for a  randomized algorithm um  what  yes or no or what do you think  um  so let's launch a poll on that  and see what your  opinion about using pseudorandom number  generators instead of truly random true  randomness for our algorithms  i'll give you a few seconds a  minute to  weigh in on that  okay  we're gonna close this down everybody's  participated who wants to uh one two  three  um  [Music]  okay  yeah i think probably the the best  answer is a  um let's take a look at some there are a  couple of answers here that  really that don't make um that aren't as  good i would say uh b  well you know usually people think of  random pseudorandom generators as pretty  fast procedures um they're not not that  interesting otherwise so  um i wouldn't say that b is a good  choice because  you know they're they're usually pretty  quick to implement  um  [Music]  c is  a worse choice even  because uh turing machines can do  anything that any other algorithm can do  so certainly if there is such a thing as  a pseudorandom number generator and  there is then you could uh implement it  on a touring machine um  d is kind of an interesting answer uh  because you're saying well that would  imply that p equals bpp if you could  actually simulate  uh randomness with a deterministic  procedure  um  but in fact the reason i i would  not choose d is because it's perfectly  conceivable that p  does equal bpp we don't know that p is  different from bpp so it's conceivable  that they're equal  and in fact i think if you if you polled  most complexity theorists  they would  most most people in my field would would  believe that p does equal bpp  just for this very reason  that if you had sufficiently good  pseudo-random number generators  you could actually eliminate the  probabilism in these probabilistic  computations you could just run them on  the pseudorandom number generator and in  fact there is some theory around that um  that has been developed um  but at the present time we do not know  how to prove that there are pseudorandom  gen number generators and it has some  actually there's actually in some line  of this research has some connection  with the p versus np problem but we  don't know how to prove that there are  sufficiently good pseudo-random number  generators that would would allow you to  run them on a probabilistic algorithm  and have a guaranteed behavior which is  as good as running truly random numbers  into the um  probabilistic  algorithm and so  the the answer that i would pick would  be a that you could use it sure you  might get the right answer but it's not  guaranteed we just don't know how to do  the analysis for the pseudorandom number  generators and you know if you had such  you had ones that were good enough they  would show p is equal to vpp  but that might be in fact the correct um  that might actually be true  okay  so let's continue on um  and uh  remember now branching programs you know  we had these uh kind of networks um of  nodes and edges  um and there was a procedure you know  we'll see a couple of examples again  some of the ones that we had from before  uh where you um  you know you have branching programs  look like this  and you know you you have a bunch of  query nodes you look at the settings of  the variables to decide whether to go  down a zero edge or a one edge and  eventually you're going to end up at an  output node and that's going to be the  output of the branching program and in  such a way these branching programs  define boolean functions  from the settings of the input variables  to a 001 output  now you might have two branching  programs and wonder whether they're  computing the same boolean function or  not and testing that is a co np complete  problem as you are asked to show on your  homework um  uh  now  if the branching program however has a  restriction namely that it's  not allowed to ask to query the same  variable more than once on a path  then  with that restriction you can  we call it a read once branching program  and then the situation for testing  equivalence seems to be different um in  fact we can give a bpp algorithm for  testing the equivalence of read once  branching programs even though such a  thing is unlikely to be the case for  general branching programs because of  the co-np completeness  okay so i hope you're comfortable and  with me on all that reasoning  um  okay  um  [Music]  let's uh  so  um  [Music]  all right so the idea for proving this  is what we're going to do is we're going  to take the two branching programs and  run them on a randomly selected input  but as we observed last time if you just  run them on a randomly randomly selected  boolean input where we assign the  variables zeros and ones  then that doesn't give you the right  answer with high probability  because the two branching programs  might be  different computing different boolean  functions but they differ only on a  single input setting  and then  just picking them at random you're not  going to have a very high probability of  finding that one place of difference  so instead what we're going to do  is define a way to run these branching  programs on non-boolean inputs where the  variables are set to values other than 0  and 1.  2 3 7 22  and make sense of that  and then argue that by running the two  branching programs on a randomly  selected non-boolean input  that that's very high probability of  giving you the right answer  um  so somehow by expanding the domain of  possibilities you're going to uh better  your chance of getting the right answer  very significantly  okay so even though these two branching  programs might agree on all of the  on almost all of the boolean  inputs  we're going to show that by  doing this arithmetization  so this is the method um  uh  they're going to if they're really not  equivalent they're going to differ  almost all of the time on the expanded  domain  okay now we have to prove  so that there is uh um  you know that's where today's work is  going to be  okay  so um  why don't we just stop  and  make sure we're all  together on this  i can take any questions um  i'll also review how the arithmetization  goes  but that i'll do that next  so are we all okay  on this  um  good so let's move on  so in order to move toward understanding  what it means to run the branching  programs on these non-boolean values  we're going to have to uh  uh get a somewhat different perspective  on the computation of a branching  program  so the standard com perspective is that  you know you um  you take your setting your assignment to  the input which is  0 1 1 for x1 x2 and x3  and use that to follow an execution path  through the machine  so you know x1 is 0 x2 is 1 x3 is one  the output is one  okay as i've indicated in yellow  um  this other perspective says well let's  let's we're gonna by operate by labeling  the nodes and edges of the machine  and  that's going to have a very direct  correspondence with the execution path  perspective so we're going to label all  the nodes and edges on the path with a 1  as indicated in yellow  and all the nodes and edges that are not  on the path all the other nodes and  edges are going to be labeled zero  okay  so by following the ones  it's like the breadcrumbs you know and  hansel and gretel these are the this is  the path you need to follow  to get uh through uh the machine  the zeros are the places where you don't  go  um  okay so the output label here  the the output is going to be the label  of the one  output node  whatever you're labeling that because if  it's a one that means the path went to  the one and if it's a zero that means  the path didn't go to the one who went  to the zero  okay so just by looking at this value  you can see what the output of the  machine is  all right so  um  let's  describe a different way of defining  this labeling without just looking at  the path  but it's going to capture exactly the  same thing  um  so we're going to say  whenever you have  you know if if you've already labeled a  node  i'll tell you how to label the two edges  that emanate from that node  um  [Music]  i'm going to label the one edge  a and  the query variable  why is that  well  a is going to be either a zero or a one  and it's going to tell us whether or not  the path  entered that node  so if it's a 1 it entered that node at 0  it didn't enter that node the only way  it's going to go down that this branch  here  is if it did enter  the x i node if it didn't enter the x i  know there's no way it can go down this  this branch so we're going to and that  value  so it's so the only way you can go down  this branch is if it went through that  node so that's the a the value of a  and  the x i is a one that's why we say a and  x i so you really have to understand  this little expression here in order you  don't understand that  you're toast  okay so you better understand this so we  can move forward i'm happy to take a  question these are the simplest  questions or sometimes the most valuable  if you understand why i'm labeling it  this way shoot me a chat  um  okay now the other brand uh other branch  i'm going to say well i'm only going to  go down this edge  if well a is true so i did go through  that node  and  x i is false so this is going to be a  and the complement of x i  all right so that's how i'm going to  label these this is another way  giving these expressions for labeling  these edges  based on the label of that node and  similarly in order to complete the  picture i've got to tell you how to  label the nodes based on the edges that  are coming into it  so if i know that i have a1 a2 and a3  which tell me  the status in terms of the path of  whether the path  went through any of these edges  well i know that it's going to go to  that node if it went it's the or of  these values  if the path went through here or it went  through there or went through there then  it's going to go through that note  that's why the or is the right thing to  say  so this gives me another way of  constructing the labeling over here  without even talking about paths  but i'm just as i describe it i argue  that it's going to get the same result  all right  so there's a question can we quickly say  again why we can't do that on boolean  i'm not sure i understand the question  so please send it to me again but this  is right now everything is boolean we  haven't done arithmetically anything yet  um  and the reason why we can't just live in  the boolean world is that  you know just by taking boolean values  of boolean assignments here we don't  have a high enough probability of  catching a difference between the two  machines  all right so let's continue um  all right so now i'm going to talk about  how to we're going to extend this  to the non-boolean case  okay using the arithmetic  arithmetization method so first of all  arithmetization is a simulation of and  and or with plus and times  such that if i think about  um  [Music]  true as a one and false as a zero  this is going to give me a faithful  simulation it's going to do the right  thing um it's going to compute exactly  the same values that we expect  so like a and b  well  um  times works just like and  um  and um right you know it does for  for one and zero  as true and false times exactly works  like and  um  and  negation is one minus  and or is going to be the sum  minus the product  and then these just give you the right  values  um a a or b  if you just calculate it out  by plugging in ones or zeros you get the  right answer just by using using this  arithmetic  um  so  now what we're going to do  instead of using the boolean labeling  we'll just use the arithmetical labeling  but it's going to compute exactly the  same thing because the arithmetic  simulates the boolean  um  so uh we always go through the start  nodes so there's no question about  labeling the very start node with a one  but now i'm going to give expressions  just like the boolean expressions but  now they're going to use plus and times  instead of ands and ors um  so let's just see remember what we  defined before we had a and  x i for this edge  i'm going to replace that what is and we  just look up here in our table in our in  our translation table  and becomes times  so we're going to replace that with a  times x i  and it's going to work exactly the same  but you know the difference is that  this makes sense  even when we have non-boolean values  times and plus are defined  for non-boolean values  um  where zans and or not  so what what goes down on this edge um  well this was  a and the complement of x i as you  remember  uh so we're going to that's going to  become a times 1 minus x i  and then similarly  we had or over here and here's a little  bit of a trick but that's going to be  important for the for the for the  analysis that we're going to do  instead of using the recipe for um or  in terms of plus and times we're going  to have something a little simpler it's  just going to be the song  and the reason why that works it's good  to understand  is that  because of the acyclic nature of the  branching programs at most one of these  edges can have a path through it  so this is a kind of very special ore  um  sometimes called a disjoint or  you're not allowed to have more than one  of the values be one because that never  happens in when you have an acyclical  graph you can never have the path coming  down this way and then again coming down  that way then it would be entering that  node twice had to be a cycle  so it's going to be good enough for us  and it necessary for us to represent  uh this or as as a sum  okay  so i think that's all i wanted to say on  this slide um so somebody's asking is a  possible sum of for some of these values  to be negative  yes the values that's a as it stands  right now some of these values can be  negative i haven't put any restriction  on what the values are are going to be  um so the input could be a negative  number and then you're going to just get  negative stuff happening uh in fact  there's negative there's subtractions  going on here so even with positive  numbers i think we did an example last  time i think we're gonna i'm gonna do  that example again uh of exclusive or um  where you're going to you get negative  numbers coming up that doesn't matter  but actually what we're going to end up  doing  is operate doing these calculations  modulo  some prime number q  okay um so i'm going to pick some prime  like  17 and do all the calculations mod 17.  um and the reason for doing it that way  is really because we're going to be  picking random assignments to the um  variables as our input  and it and you need to it makes it makes  the most sense to do that when you have  a finite set of possibilities to pick  them up  um so we're not going to pick like a  random integer there's infinitely many  possibilities and you could yeah you  could set up a distribution there but  that's very complicated that actually  might work i'm not sure i i haven't  actually gone through that analysis but  the typical way people do this is by  looking at what's called a finite field  so i'll talk about that in a second um  why is there at most one one among a1 a2  and a3 so um  the ones i'll say one once again but the  ones correspond to the path so this is a  one if the path went this way  just think about it the path cannot go  through a1 and  can at the same time go through a2  because that means the path went through  this node  then  how is it going to get over to a2 it's  going to go through that node twice  in an acyclic graph you cannot have a  path going through it's the same node  more than once  so you have you're going to have to  think about that  um  okay let's move on  so now we're going to talk about the  same um  uh  we're going to  we're going to look at that  non-boolean labeling applied to an  example so here is a  very simple branching program that  actually computes the exclusive or  function in the boolean world  um  so this is the labeling that we just  i just developed for you the the uh  arithmetical labeling um and we always  label the start node with one because  the path always goes through there um  and now let's look at this before we  jump ahead let's look at this edge here  remember what it is we have to apply  this rule here this is the one edge  coming out of a node that already is  labeled  so it's that that label on that node  times the x i because that if you're  thinking about that's the and captures  the end  so it's just a times x i so xor it's x1  in this case so it's going to be x1 is a  2  in our input  so it's going to be 1 which is the a  times the x one which is two so that's  just this this edge gets labeled two  now this well okay let's look at this  edge now i think that's next  this is the one minus x i it's one minus  x one so it's one minus two so we're  going to end up with a minus one one  minus times minus  one times one minus two which is minus  one so it's a minus one  now we're going to label these two nodes  using the other rule  um  so this gets a 2 because that's the sum  of the incoming edges this gets a minus  1 because that's the sum of its incoming  edges  and now we're going to look at um  uh  which order did i do this in okay it's  going to do this uh  this edge now  um zero edge which is going to be 2  times 1 minus  its variable so it's 1 minus x2 x2 is a  3  so it's 1 minus 3  it's a minus 2 so 2 times minus 2 is  minus 4.  okay i don't want to rush through  there's no point in just blabbering on  i'm just trying to do work this example  so you get the idea  okay so could you fill this out this  next one out yourself  so this is the one outgoing edge from  this x2 node  um so x2 is a x2 is labeled  uh  with two here so there's a is two  this is the one edge going out so it's  the one on this side here so it's  two times the  x2 x2 is a three so it's two times three  so this should be six  oops right here six  i'm gonna do the same thing so  x2 is labeled minus one  um  so minus one times one minus so you got  a two here  you get a minus three here just  following again the same um process  and now let's let's say what is the  label on the zero node here so you again  you take the sum of its incoming labels  so there was a two  and a six  so that's an eight  and this is a  uh  a minus three and a minus four coming in  so it's a minus seven what's the output  the output is minus seven because that's  the label on the one node  okay output is minus seven  okay  so this is going to be our  way of  defining  the  output  of a branching program when it has a  non-boolean setting on its inputs  if you had the boolean setting on its  inputs  and you followed this procedure what  would you get out  you would get the same answer that you  would get by following the path  because the  arithmetical simulation is a faithful  simulation of the ands and ors and the  ends and auras capture exactly when the  what the path does  so this is a strict extension  of the operation of the branching  program into a new realm these  non-boolean values on the old realm it  behaves just as it would or  did originally and that's critical  to understand that  um  yeah somebody's asking would the final  value of the zero state also be the same  sure  in the ba in the in the boolean case  if we follow this in the boolean cases  all of the labels would be exactly what  they were  in the bool you know  you know they would just be the zeros  and ones that we had from before  so does this exactly mimic xor if you  take this all mod  2  hmm  i don't know i'd have to think about  that i don't think that that's essential  um in this case it might behave  correctly if you take the answer mod 2  for xor but the xor is going to be very  special i'm not sure that might happen  to be true i'd have to think about it  for a second but um  i'm not sure that's in  relevant  um  okay this is a good question if i picked  a different branching program that also  implements xor  um so it would be an equivalent  branching program  would it behave the same way  on the non-boolean values  and the answer to that is  yes and no  you understand the question it's a very  good question and it's actually we're  going to prove this  you know  in the analysis  it's going to be easy to prove because i  gave you both possibilities yes and no  but let me tell you what i mean by that  so um so you understand the question  suppose i had a different branching  program i'm not sure you can come up  with a different branching program but  let's make let's let's let's say you  could you have a different branching  program yeah sure you can come you can  do the variables in a different order  for example so you suppose you come up  with a different branching program that  gives you  xor  and now you plugged in  two and three  would i always get the same value out  yes  if that other branching program was read  once  no  not necessarily if it's not read once  and that's why read once is critical  as we will prove  for two read once branching programs if  they behave the same way  on the boolean values  they behave the same way  even on the non-boolean values  that's not necessarily true if the  branching programs are not read once  okay so let's we will see that we will  prove that  and use that that's going to be  important  okay so here is the algorithm sketch  which is kind of a little bit even sort  of suggested by that uh that that  that very good question um  uh  so what we're going to do is we want to  take the um the two branching programs  for with that we're trying to test if  they're equivalent  we're going to pick a random non-boolean  input assignment  so set the values here  chosen from the field but we'll get  there  you know  a random value for x1 a random value for  x2 and so on these could be numbers like  17 and 25 and 23 and so on  and then run  using this process run the branching  programs  to evaluate them on that non-boolean  assignment  if they  disagree then we're going to know that  the two branching programs were not  equivalent even on the non-boolean case  even on the bullying case did i say that  wrong so if they disagree even on the  non-bullying case they have to be an  equivalent even in the boolean case  not obvious  but if they agree  and if they agree  then it's not a guarantee that they're  equivalent but it's going to be very  strong evidence that they're that  they're equivalent  okay so that's where the probabilistic  nature is going to come in  okay so we're going to prove that  so first we have to develop an algebraic  fact  um  and that involves polynomials  this is this is a simple thing that i  think many of you have run into already  perhaps even you know in high school um  uh i'm not going to prove the algebraic  facts but i'm going to state them and  actually the proofs are not hard they're  in the textbook um  okay so let's suppose we have a  polynomial  of degree d here it happens to have b so  you know some kind there's a bunch of  constants these a's are constants  x is the variable of the polynomial  and uni i presume you've seen  polynomials written out like this  and so if you have if you assign x to  some value some constant value z  um  and the polynomial value evaluates to  zero  we often call that a root of the  polynomial  um so the polynomial these are the  places where the polynomial evaluates to  zero and i've kind of shown over here  those are the roots  it's not hard to show  that a low degree polynomial cannot have  lots of roots  basically if the polynomial has degree  at most d it can have at most d roots as  long as the polynomial itself is not the  everywhere zero  polynomial because obviously then  everything is a root  oops typo thank you should be d minus  two over there  uh  pretend that's a two  uh  all right so um  all right so if we have a low degree  polynomial let's not get ahead of  ourselves below degree polynomial uh  degree did you do the polynomial of  degree of most d it has the most d roots  and that's a simple proof um the basic  idea is every time you have a if you  have a root of the polynomial you know  so if um  uh  if setting x equal to five  gives you a root of the polynomial um  you know it's a zero of the polynomial  then x you can easily see that x minus  five is a factor of the polynomial you  can divide out x minus you can divide  out  by x you can divide by x minus five and  you get a new polynomial of degree one  less  and you can just uh which is going by by  induction has a one fewer uh root um so  you can just divide out by the roots  basically it's very straightforward  uh  um  uh  okay so now um other very important  thing  uh  if i have two polynomials  that are both low degree they cannot  agree  on very many places  that follows right  from the from what i just proved above  because what i'll do is i take those two  polynomials and i look at the difference  which is also a low degree polynomial  every time there's an agreement between  those two original polynomials there's a  zero of the difference polynomial  and because that difference polynomial  cannot have too many zeros the two  original polynomials cannot have too  many agreements  okay  so the corollary is that if x and y are  both polynomials of degree of most d  and they're not the same polynomial  because then the difference would be the  zero polynomial  then the number of places where they  equal is the most d  okay so the proof is just letting p be  the difference of p one and p two  very standard kind of a trick  okay now the the above is going to hold  for any field so we're gonna be a field  is just a set with plus and times with  the familiar  um properties  the distributive law and so on and  identities and all the stuff that you  would expect plus some time to have in  in the normal world  um and so we're going to talk about  finite field  that has only q elements  um  that has exactly two elements where q is  a prime number so it turns out and i'm  not going to prove all this but it's  pretty simple stuff  that if you just take the numbers from  one to q minus one  from zero to q minus one um they uh and  use plus and times mod q  that that does has all the right  properties to be a field  okay so we just think of it it's just  modular arithmetic  mod some prime q  okay and then it's we can in a natural  way pick  a random  assignment  to  a variable from the field because it's  just choosing from among queue  possibilities  so uh  uh  yeah so the  yeah so getting a question here the the  the the coefficients of the polynomial  and the assignment to the variables  they're all going to come from this  field  so everything's going to be we're all  going everything is going to be  operating uh  in this field don't get don't let that  throw you off or you're just your  ordinary intuition about the way  arithmetic works is going to be just  fine  um  okay  now  if we have  ah but this is important here from the  perspective of thinking about this  probabilistically so i'm going to  rethink about this polynomial lemma  which says there are not too many  roots  um in terms of the probability of  picking an element of the field what are  the changes that it happens to be a root  so if you have a low degree polynomial  and you pick a random value in the field  what's the probability that you got a  root  well it's just the number of roots  divided by the size of the field  so if you have this really big field and  you have this low degree polynomial it's  going to be pretty unlikely that that  you're going to end up picking one of  the zeros  one of the roots  uh just at random  that's all that's this is saying  so there's the most d roots out of the q  possibilities  and the last thing i'm going to  introduce here is the multivariable  version of this  um which is called  perhaps somewhat unfairly but uh it's  called the schwarz zipple lemma  though in various forms it had been um  known  prior to their work um  in some cases  you know the literature actually goes  back  a long ways but anyway the  this is called the short zipline it  doesn't really matter um uh  except to the people whose  um  uh  credit is being denied but  that's not one of us uh so anyway the  schwarzepilema  says that if you have now a polynomial  in several variables which is not the  zero polynomial  where each variable has low degree so if  i say if has degree at most d in each x  i  so each variable is going to have at  most an exponent of d appearing in that  polynomial  and now if we pick  random values  to assign to all of those m variables  from the field  the proper probability that we ended up  with a root  and we ended up with a zero  is something you can bound  so it's m times d so the number of  variables times this maximum degree  divided by the size of the field  and this is going to come up later for  us  and this is another  fairly simple proof  a little bit more sophisticated than the  one that we had above and in fact she  uses that one as a um  as a lemma to prove this theorem  okay so we're going to uh not going to  prove any of that but i refer you to the  book if you're curious  um  uh  so  um  yeah so like a couple of good questions  here  what happens if these values are bigger  than q for example then then then it's  not telling you anything if d is bigger  than q m is bigger than q or the product  is bigger than q  then  then you learn nothing  uh from this lemma from this theorem  so  typically what the in applications  you're going to pick a larger you're  going to have the flexibility you get to  choose q to be in something that you  want so we're going to pick the field to  be big enough  so that the m and the so the m and d are  going to be relatively small  um in fact d is going to end up being  one as we will see  um and m is the number of variables so  we're going to pick q which is going to  be substantially larger than the number  of variables  and how is the degree defined in  multi-variable polynomials you know the  m if the polynomial has  you know  x y squared plus  3 x to the fifth y squared z you just  you just pull out each variable  separately  and you look at the maximum degree of  that variable so the x in that case had  had a degree five  appearance the y had a degree two  appearance so you take the maximum over  all of the variables of the degree of  that variable and that's going to be the  bound on the degree of the polynomial  polynomial  so in fact in our case d is going to be  1.  so all of the variable there's not going  to be any exponents on anything  everything's going to be exponent one  um  so q  is is q related to the number we choose  for the mod yeah q is the number we're  choosing for the mod  we're doing everything mod q  so all of the arithmet all the  arithmetic is going to operate mod q  and that's the size of the field  that we're going to pick  okay so i think we're here at the break  um happy to take some more questions um  but why don't we just start that off  uh  and uh  i will see you in five minutes but in  the meantime happy to shoot me questions  okay so what happens if we use boolean  assignments in the xor example  um  so the x you know would that work  um to be able to check  uh uh agreement  um  it would it was so it's hard to it's  hard to  make an argument based on just a single  example  i think the better thing would be  to look at  you know two um  um  branching programs that just differ in a  single place so i can even suggest to  you can make a branching program that  always outputs um  true  it doesn't even read its variables or if  it reads them and they always go to the  same place and then ends up always at  the one output so imagine a branching  program that always outputs one no  matter what the assignments to the  variables are and you can you can easily  make such a thing  and then you make another branching  program that computes the or function  so it reads every variable  and  it's going to be  1 if any one of those variables is set  to 1.  okay so  the only time the or function is 0 is if  everything is set to zero  but  now if you're trying to randomly check  whether the always one function  is equal to the or function  of course without knowing in advance  what they are because that's cheating  you're just given these two functions  and you want to know these two branching  programs  and you want to know are they computing  the same thing or not  and by this procedure of randomly  sampling  there you're always going to get these  branching programs both to say one  unless you just happen to pick the  random assignment of everything set to  zero  and that's very unlikely that you're  gonna pick that you know if you imagine  you have your branching program has a  hundred variables in it it's only two to  the minus 100 chance that you're going  to set them all to zero  randomly  and so you're almost you go you're  extremely unlikely to find the one place  of difference if there's only a single  place if there's lots of places of  difference then you  you know it's not not so bad but if the  number the fraction of differences is  low  you're going to have to do a lot of  samples in order to find that it won't  be in you know possibly exponentially  many samples and then you won't run a  polynomial time okay so let me just see  if there's other questions here um  why can we accept  going back to the boolean labeling side  why can we accept that b1 equals b2 if  b1 and b2 agree on  only on just one input assignment no we  didn't say that  um all right i'll go back there boolean  assignment  is this i'm not sure which one you mean  is this one what do you mean  i don't know  boolean labeling site must be it  why do we  non-boolean labeling okay  so what  no  i see what you're saying you're saying  about this here we're just going to pick  one random assignment  and if they agree on that one random  case  then we will say um accept  because you might think well we should  take to a whole bunch of samples that's  a good question  um but in fact we're going to arrange  the probability such that if the two  things are not equivalent  then you're  it's going to be uh diff the value the  values will be different almost  everywhere a large number of places so  just picking one and having them agree  is going to be strong enough evidence  that you're still going to accept and  you'll have a still a low probability of  getting an error  you'd have to see the analysis  um  are we assuming the roots of the  polynomials are integers no  don't forget when operating over a field  here so it's not even talking about  integers doesn't totally make sense but  it doesn't really matter um we're not  assuming that the bound still  uh  the bounds still oh i should have taken  this away um  the bound still works  um uh  the bound still works  even even if we have non-integers  okay i'm not sure if i'm being helpful  here why don't we just move on  um but we're not assuming that these are  integers because the bound doesn't  doesn't matter if  it says there's at most five roots  including the reals there's still going  to be five roots including the integers  all right so let's continue  um  [Music]  good  all right  so now  everybody's back i hope  let's talk about  moving forward here  um we want to basically what where we're  going with this is we want to  analyze  the algorithm  which picks a random non-boolean input  and evaluates the two branching programs  and in order to do that  we're going to look a little bit more  carefully  at the what happens  uh when we  arithmetize the branching program  and um we run it on these non-boolean  values and so what i'm going to do  is  take these take this branching program  let's say this is the same  xor exclusive or branching program but  instead of  labeling it  as we did before by setting x1 to 2 and  x2 to 3  i'm going to leave x1 and x2 variables  and just do a symbolic execution  so i'm going to label these things  just leaving x1 and x2 as variables  okay so let's just see what we get if we  do that  so  remember we assigned this to be one  now  this edge here  is  you know here's the rule it's a which is  1  times  x1 so this should be  without knowing what the value of x1 is  leaving it as a variable we're just  going to put down x1 over here  over here what goes over there  well it's 1 times 1 minus x1  just 1 minus x1  now we're going to add things up as we  did before  and now what happens for example i think  i have this edge coming next let's look  at this edge the one edge coming out  so that's the label now is x1 on this  node  and the out this is the edge this is the  one edge coming out so you multiply by  the um by the value of x2 that we're  leaving it as a variable so you're just  going to multiply it by x2 and so we're  just going to get x1 times x2  x1 x2  on this edge now what happens on this  edge  so this x1 times think with me  um  times  1 minus x2  and similarly over here we have 1 minus  x1 on this node  so i think on this  on the one edge coming out it's 1 minus  x1 now times x2  because that's this rule again  1 minus x1 times x2 and this is going to  be 1 minus x1 times 1 minus x2  now we're going to add this up  for the zero  node  so we have this this value 1 minus x1 1  minus x2 plus x1 x2  and on this node here we're going to um  add these two values up  so 1 minus x1 times x2 plus x1 times 1  minus x2 and that's the output now  expressed symbolically  now you could plug things in and you're  going to get the same value out as you  did before  um  but let's leave it as a polynomial  for now uh because that's going to um  [Music]  help us and analyze this  so now notice the form of this  polynomial is something special  um  what happens is it's gonna look like a  bunch of products  of  x i's and one minus x i's  added up so it's a sum of products of  that form  so each row here is a product  of x i's and one minus x i's  and then those rows are all added  together  i claim that's going to be the form  of this polynomial you see this already  has that form  um  and the reason for that is  every time  you go to a node you're just adding  things up so that's just going to be  like adding adding up more rows  and every time you go down to through an  edge you're multiplying what you have so  far either by  an x i or a one minus x i  so you're just accumulating products of  x i's and one minus x i's and you're  just adding them up so this is what that  polynomial is going to look like  okay  now let's look a little bit more  carefully  at  the form of this  so for one thing  could we have  um  higher powers here  like a x2 cubed  could that happen  so because you know when i say it's  products of x i's and one minus x i's  maybe there's going to be some x xi's  that appear several times in the product  well that cannot happen  why  read one sprint it's a read once  branching program  so every time you multiply by an x i  or one minus x i  you're never going to do that again  because  doing that would imply you're querying  that variable more than once  so this can't happen  okay so i crossed that and this is sort  of appeared off to the side here but  yeah i'm crossing that out that does not  happen  another thing  that  is part of that's worthy  that's going to be helpful to to notice  about this  this polynomial and by the way maybe i'm  i'm being confusing here this is i'm  supposed to be  representing sort of as a generic form  of the polynomial this is not some  particularly i shouldn't have said this  at the beginning this is not some  particular polynomial that came from  anything i'm just trying to describe  what the general form of the polynomial  looks like  just as an illustration  so you know it's so this polynomial is 1  minus x1 times x2 times 1 minus x3 times  x4  and so on  and and adding up a bunch of rows like  this i'm just saying this is what the  polynomial will look like  for maybe some branching program so  every branching programs that are going  to have some polynomial that looks sort  of like this  um  and what i'm also going to say  what for convenience now i want to say  that each row is going to have every  single variable up here  either as an  x i or as a 1 minus x i  so in order to get that i need to make a  further minor assumption about the  branching program that it's a read  exactly once  currently when i say read once it can it  can avoid reading some variables on some  branches because i could read it most  once but now i want to say that every  variable gets read exactly one time on  every branch  and what that's going to mean is that  every row is going to contain every  variable  either  as an x i or as a 1 minus x i  we can eliminate that extra assumption  easily and i'm going to leave that as an  exercise to you it's not very hard to do  um so i think if you follow me you can  see  and you play with it for a minute or two  you'll see that it doesn't really matter  uh you can either you  but i think just for the for the first  time through this  let's assume that every row has every  variable  okay so important to understand um  okay so  so the this is the output polynomial of  this branching program so  let's look further more at this  polynomial  and understand  the rows  let's take one one row  out of this polynomial  and to understand its um  uh  um  understand what it represents  so one row here it's a product of a  bunch of things  part of a bunch of variables either  variables or one minus the variables  now  notice  let's think about this in the boolean  setting first of all  so in the boolean setting  each of these variables are going to be  zeros and ones  and the one minus the variables are also  going to be zeros and ones so it's going  to be a product of zeros and ones  if there's a zero that appears in that  product  that product is going to be a zero  because zero times anything is a zero  so the only way that product cannot be  um zero  is if all of those  uh  values are once in the boolean case  so  uh that means that  x1  well let's  let's let let's look at the second row  so x1 had to be a one x2 had to be a one  x3 had to be a one x4 had to be a zero  um in order to continue the product of  ones  and so  on  so in fact there's only a single boolean  assignment  to these variables which make that row  one  every other assignment to those  variables makes that row zero  saying that another way  these row each of these rows corresponds  to one of the rows  of the  truth table for the boolean function  where  where the truth table is true gives a  true value gives a one value for the  function on that row  okay so i i hope you've all familiar  with the notion of a truth table of a  boolean function you just write down the  boolean function  every possible assignment to the boolean  function and you write down  one or zero or true or false for what  the value of that function is so just a  tabular representation of the boolean  function it's called the truth table  this thing here gives you all of the  ones all of the rows that are one in  that boolean function that's what this  polynomial gives you  okay so i think we're  at a pause point  for  um this slide and i  um  it's deathly silent on the chat  so i was feeling that that was a  that went down that went down rough  uh for for you um it's it's we're gonna  it's important to understand the form of  this polynomial  uh  here um  you know that it's corresponds to the  truth table of the boolean function so  each one of these rows is only going to  be on again thinking boolean now  each one of these rows is only going to  be one  on that on an assignment which makes the  function one  um  one of the one of the assignments that  makes the function one  okay and somebody says and similarly for  the expression for the zero node yeah  the zero node which i'm not focusing on  but yeah the zero node would be all of  the false rows of the truth table  but the one node the polynomial for the  one node are all of the true rows  correspond to all of the true rows in  the in the in the um  in the um  uh  in the  in the  function of the branching the function  that branching program computes now  the reason why this is let me just tell  you where we're going  is it possible to have two rows that are  the same  if you think about how the rows are  being produced  no you can't have two rows of that are  going to be the same um  uh  because for one thing  you have to think about what what this  looks like when the boolean case if you  have two rows of the same that means  this thing is going to have an output  which is non-boolean  um  because you're going to end up with a 2  coming out that way  by adding those rows together  they can never happen and if you just  look at the way it's constructed you're  never going to because every time you  have a branching  you know one way is a an x one an x i  the other one is one minus x side so  every time you're branching there every  not every time there's a node they're  different um so you're never going to  have two rows that are going to be the  same but  let me tell you the importance of  connecting up  the this polynomial with the truth table  um because that tells us  that if the two functions of the two  branching programs that we started off  with  agree in their boolean values  then the two polynomials are going to be  the same  um  [Music]  you know because if the two branching  programs  uh have the same  boolean function so they're equivalent  then the um  truth tables will be the same and  therefore these polynomials will be the  same  so that and therefore they will behave  the same way on all  non-boolean values because they're the  same polynomial  so i'm sort of getting ahead of myself  but that's where that's what we're going  to argue that's why it's important to  understand the connection with the truth  table  because it sort of builds on the  um  under you know understanding something  about how this thing behaves in the  boolean case is going to give us  information about how it behaves in the  non-boolean case  okay  let's continue here then  um  okay so this is i think is this my  um  yeah this is essentially the last slide  okay um  we're going to spend some time on this  one so here's the algorithm  we are going to take our two branching  programs  the variables are x1 to xm  first of all we're going to find a prime  which is at least  three times  m the number of variables  m is not a very big number  um it's just a number of variables so  finding a prime that's bigger than that  is straightforward we're not talking  about huge primes here we're talking  very modest size primes so you can just  you know just try them  even trial and error is going to be good  enough um  now  that's going to be the  the size of the field it's going to be  field of size q um  and now we're going to pick  a non-boolean assignment to the  variables  and  okay we're going to pick a non-boolean  assignment to the variables  and  [Music]  evaluate the two branching programs on  that non-boolean assignment using the  arithmetization  if they agree  then we'll accept if they don't agree  then we're going to reject  okay now we have to argue that this  works  so we're going to first of all  arithmetize these two branching programs  i'm going to get the two polynomials  they each have the form that as i  described  um so a bunch of rows that correspond to  the truth tables of those two respective  branching programs  okay first claim  that if the branching programs were  equivalent so they compute the same  boolean function  then  the two polynomials agree everywhere  um  so then the two branching programs are  going to get the same value  um  on every non-boolean case as well as on  the boolean cases so they agree on the  boolean that means they always agree  even on the non-bully and i kind of  argue that already  but okay the other point is that if the  two branching programs  are not equivalent so they  differ  at some boolean value  now picking a random value to eval for  the polynomial evaluation you're going  to have only a one-third chance that  they're going to agree  so a small chance  all right so now um  let's prove these two facts the first  one i already kind of argued if the two  branching programs agree on all the  boolean values  um  then their functions have the same truth  table  so then the polynomials are identical  because the polynomials correspond to  the truth table  and so therefore they always agree  even on the non-boolean values  okay so  that means that the probability that if  you evaluate the two polynomials on a  random place  the it's good  whether you'll be equal that's a  certainty  because in fact in this case p1 and p2  are the same polynomial  now for two  if the  branching programs differ somewhere even  in one place  well you know the polynomials could not  be the same  they have to be different polynomials  because the polynomials include the  behavior of the  in the boolean case as well as all the  breadth of the field  um  so the polynomials have to be the same  and now we're going to apply the short  zipple lemma  we have two  different polynomials  they can only agree in a relatively  small number of places  so that says that  from the short simple theorem then um  the probability that p1 and p2 agree at  this random location is at most this  value that we had from before  the degree times the number of variables  divided by the size of the field  the degree is one  and the field is at least three times  uh the number of variables in size so  that means um  you get this inequality here and so  therefore the probability is it is a  third  and most a third  and that's good enough  um this is the probability that you get  the wrong answer it's going to be at  most one-third  so you're going to get the right answer  with at least two-thirds probability so  even just doing a single sample point is  going to be enough to to give a bpp  algorithm  all right so that's  um  what i have is are a couple of check-ins  here  uh now for you  um  [Music]  whoops  somehow this  better take me out of here um  uh  all right  so this is this is a little hard but  let's see how you do on it  suppose the branching program is  uh  well maybe i sort of a little bit  discussed this already that's okay the  branching problems will not read once  the polynomials might have exponents  bigger than one so where would the proof  fail  um would they fail the point where  b1 and b equivalent to b2 implies that  they agree on all boolean inputs so  that's sort of this  um the first step here  or  um  is it that agreeing on all boolean  inputs implies that the polynomials are  the same  or would it be that having the two  polynomials being equal implies that  they always agree  so those are the three sort of steps in  the  part one of the the proof of part one  so let's see uh what do you think there  in a couple questions about picking the  prime number  um  the this uh the prime number here is uh  it's this is a very small prime number  um  you could even represent that prime  prime number in unary  within the amount of time we have  because don't forget this is a prime  number whose magnitude  it is at most the number of variables so  you can write that prime number in unary  and um  uh  finding the prime and testing primality  of testing whether the number is prime  is something that you can do even with  this brute force algorithm  and it would be good enough  um so there's not there's no you don't  have to do anything fancy about testing  primality um in this case  okay so why does it have to be prime  you need it to be prime in order for the  for the um  for it to be a field  so this is just the algebra part  um  if you did not have a prime number then  then some of the field properties don't  work and you may no longer get the um  the fact that the polynomial has a small  number of roots  um  so that  that's all i can say about that is the  polynomial like a hash function for the  branching program are the equal if they  are the same but sometimes the values  also equal if the programs are different  that's an interesting idea so is that  is a polynomial acting like a hash  function i think it's there is something  to what you're saying but i think it's  actually in the other direction um  uh  it's it's actually more it's related to  a hash function but it's actually acting  more like an error correcting code  let's let's save that for later it's a  very good point it's a very good  question maybe we can talk about it  after if you remind me  um  okay let's let's end this poll here  um  should c be  if  having these two agree implies well i  mean  the question is should should we change  p1 and p2 always agree to b1 and b2  always agree well b1 and b2 are behaving  exactly the way b p1 and p2 behave so  i'm not sure it matter really matters  um  uh  too much time on this chat on this poll  here let's end this poll  oh sharing results  um  yes so the correct answer is b  um  that agreeing on all boolean inputs  implies that they're equal  um  the other two are follow kind of  immediately they're still true  but  if it's not read once even though they  agree and all the boolean inputs they  won't necessarily agree  as polynomials  um for one thing like if you just take  the two polynomials  x1 squared and x1  they agree on all the boolean inputs but  they're not the same um  they agree in the boolean world because  zero squared is zero and one squared is  one but they they're not the same  polynomial  okay uh let's move on actually i have  another  uh check in on the same slide here  um  and this is actually answering a  question that we i got in the chat if  p102 where x  you know  how big how big are these polynomials  are they these  look like they could be big if they're  exponentially large would that be a  problem for the time complexity  so  pick a or b  here we're running out of time here so  why don't i uh  not say too much and just let you go  with it  okay  i'm gonna close this down  all in  well  my god  by one point  this is a  this is  uh  this is like georgia here um but do a  recount um  in fact  b was is correct  uh  by it by by a hair here um they are not  polynomial in size the truth tables can  be very large but we don't have to  as we did with the um  branching program for  um  the exclusive or case you don't actually  write down the polynomials to evaluate  them you can evaluate them as you're  going along the polynomials are huge  but you don't have to write down the  polynomials to evaluate them that's only  part of the proof  you don't have to re either the the  algorithm doesn't have to deal with the  polynomials itself  so  maybe a  good uh  good to think about this  that somebody says did i invent this  proof no i did not invent this proof  it's a wonderful proof but it's not mine  i would love to have been get take the  credit for it  okay so um  why don't we uh  wrap this up  so you know can we just is there some  way to simplify the polynomials as we're  going along so that we we don't end up  with them being too big um  and so that we can then just look at the  polynomials not that i know of  um  uh i i i think the polynomials are  really going to be big  and so there's not going to be any way  to um  uh to to view it just as pawn if you  could it would be fantastic because that  would give you a deterministic algorithm  um  i think the only way that people know  how to do this is terms of you know  random inputs the polynomial to a  polynomial which is too big to write  down if you could write it down and just  analyze the polynomials  you'd have a huge huge result there  um  okay  i'm glad  people like this proof that's good  um  how many actual quantities are there in  the formula i'm not sure what that means  what formula  um i mean the polynomial is huge  uh the number of different polynomials  well  i i guess i don't understand the  question what motivates the idea of  arithmetization  what would make somebody think of this  i'm not sure  actually  um  but we're going to use it even in a more  remarkable way  uh in the in the last two lectures of  the course  so stay tuned  i mean this is sort of a clever but  seems very specialized but the um  the next part  where we're going to next application  we're going to use this to analyze  satisfiability which is much more  general kind of a situation and that's  that that i think is especially  remarkable  um  um  for the polynomials  p1 and p2  uh it's only a polynomial number quantum  i don't think so because well it depends  on the size of the field the size of the  no it's going to be something like m to  the n m power right  um so those are the number of possible  inputs  so it's in each each each field element  is  is  has  3m possibilities roughly and there are m  field elements so it's m to the 3m  different possible  inputs  that you're picking at random  so anyway i'm going to shut this down  and move over to the  to the others the office hours zoom  feel free to join me there otherwise i  will see you all on thursday  take care  you
 

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312
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313
00:09:41,360 --> 00:09:43,509


314
00:09:43,509 --> 00:09:46,070

 

315
00:09:46,070 --> 00:09:47,910

 

316
00:09:47,910 --> 00:09:49,590

 

317
00:09:49,590 --> 00:09:51,670

 

318
00:09:51,670 --> 00:09:53,829

 

319
00:09:53,829 --> 00:09:56,150

 

320
00:09:56,150 --> 00:09:57,910

 

321
00:09:57,910 --> 00:09:59,990

 

322
00:09:59,990 --> 00:10:02,710

 

323
00:10:02,710 --> 00:10:04,389

 

324
00:10:04,389 --> 00:10:05,990

 

325
00:10:05,990 --> 00:10:07,350

 

326
00:10:07,350 --> 00:10:09,350

 

327
00:10:09,350 --> 00:10:10,870

 

328
00:10:10,870 --> 00:10:10,880

 

329
00:10:10,880 --> 00:10:11,260


330
00:10:11,260 --> 00:10:11,270

 

331
00:10:11,270 --> 00:10:12,949


332
00:10:12,949 --> 00:10:14,310

 

333
00:10:14,310 --> 00:10:16,389

 

334
00:10:16,389 --> 00:10:18,550

 

335
00:10:18,550 --> 00:10:20,470

 

336
00:10:20,470 --> 00:10:23,110

 

337
00:10:23,110 --> 00:10:24,630

 

338
00:10:24,630 --> 00:10:26,710

 

339
00:10:26,710 --> 00:10:29,829

 

340
00:10:29,829 --> 00:10:31,829

 

341
00:10:31,829 --> 00:10:33,670

 

342
00:10:33,670 --> 00:10:36,069

 

343
00:10:36,069 --> 00:10:37,829

 

344
00:10:37,829 --> 00:10:40,150

 

345
00:10:40,150 --> 00:10:40,160

 

346
00:10:40,160 --> 00:10:41,350


347
00:10:41,350 --> 00:10:41,360

 

348
00:10:41,360 --> 00:10:42,230


349
00:10:42,230 --> 00:10:45,190

 

350
00:10:45,190 --> 00:10:47,750

 

351
00:10:47,750 --> 00:10:49,670

 

352
00:10:49,670 --> 00:10:52,550

 

353
00:10:52,550 --> 00:10:54,870

 

354
00:10:54,870 --> 00:10:56,069

 

355
00:10:56,069 --> 00:10:59,190

 

356
00:10:59,190 --> 00:11:02,310

 

357
00:11:02,310 --> 00:11:04,310

 

358
00:11:04,310 --> 00:11:06,790

 

359
00:11:06,790 --> 00:11:09,829

 

360
00:11:09,829 --> 00:11:11,829

 

361
00:11:11,829 --> 00:11:14,230

 

362
00:11:14,230 --> 00:11:16,310

 

363
00:11:16,310 --> 00:11:18,310

 

364
00:11:18,310 --> 00:11:20,310

 

365
00:11:20,310 --> 00:11:22,230

 

366
00:11:22,230 --> 00:11:23,910

 

367
00:11:23,910 --> 00:11:27,190

 

368
00:11:27,190 --> 00:11:29,590

 

369
00:11:29,590 --> 00:11:31,990

 

370
00:11:31,990 --> 00:11:34,310

 

371
00:11:34,310 --> 00:11:36,150

 

372
00:11:36,150 --> 00:11:38,150

 

373
00:11:38,150 --> 00:11:39,509

 

374
00:11:39,509 --> 00:11:41,350

 

375
00:11:41,350 --> 00:11:43,269

 

376
00:11:43,269 --> 00:11:44,949

 

377
00:11:44,949 --> 00:11:47,670

 

378
00:11:47,670 --> 00:11:50,790

 

379
00:11:50,790 --> 00:11:52,949

 

380
00:11:52,949 --> 00:11:54,949

 

381
00:11:54,949 --> 00:11:57,269

 

382
00:11:57,269 --> 00:11:57,279

 

383
00:11:57,279 --> 00:11:58,870


384
00:11:58,870 --> 00:12:00,949

 

385
00:12:00,949 --> 00:12:02,470

 

386
00:12:02,470 --> 00:12:04,949

 

387
00:12:04,949 --> 00:12:06,389

 

388
00:12:06,389 --> 00:12:07,910

 

389
00:12:07,910 --> 00:12:10,389

 

390
00:12:10,389 --> 00:12:12,470

 

391
00:12:12,470 --> 00:12:14,389

 

392
00:12:14,389 --> 00:12:17,110

 

393
00:12:17,110 --> 00:12:20,150

 

394
00:12:20,150 --> 00:12:22,949

 

395
00:12:22,949 --> 00:12:22,959

 

396
00:12:22,959 --> 00:12:24,550


397
00:12:24,550 --> 00:12:26,710

 

398
00:12:26,710 --> 00:12:29,190

 

399
00:12:29,190 --> 00:12:31,509

 

400
00:12:31,509 --> 00:12:34,870

 

401
00:12:34,870 --> 00:12:36,550

 

402
00:12:36,550 --> 00:12:38,949

 

403
00:12:38,949 --> 00:12:40,470

 

404
00:12:40,470 --> 00:12:42,310

 

405
00:12:42,310 --> 00:12:44,230

 

406
00:12:44,230 --> 00:12:45,590

 

407
00:12:45,590 --> 00:12:47,670

 

408
00:12:47,670 --> 00:12:49,350

 

409
00:12:49,350 --> 00:12:51,590

 

410
00:12:51,590 --> 00:12:54,069

 

411
00:12:54,069 --> 00:12:56,230

 

412
00:12:56,230 --> 00:12:57,670

 

413
00:12:57,670 --> 00:13:00,150

 

414
00:13:00,150 --> 00:13:02,069

 

415
00:13:02,069 --> 00:13:03,670

 

416
00:13:03,670 --> 00:13:05,509

 

417
00:13:05,509 --> 00:13:07,750

 

418
00:13:07,750 --> 00:13:10,710

 

419
00:13:10,710 --> 00:13:12,550

 

420
00:13:12,550 --> 00:13:14,150

 

421
00:13:14,150 --> 00:13:15,910

 

422
00:13:15,910 --> 00:13:19,509

 

423
00:13:19,509 --> 00:13:21,670

 

424
00:13:21,670 --> 00:13:24,870

 

425
00:13:24,870 --> 00:13:24,880

 

426
00:13:24,880 --> 00:13:25,910


427
00:13:25,910 --> 00:13:25,920

 

428
00:13:25,920 --> 00:13:26,870


429
00:13:26,870 --> 00:13:29,190

 

430
00:13:29,190 --> 00:13:32,069

 

431
00:13:32,069 --> 00:13:35,269

 

432
00:13:35,269 --> 00:13:38,470

 

433
00:13:38,470 --> 00:13:38,480

 

434
00:13:38,480 --> 00:13:39,990


435
00:13:39,990 --> 00:13:43,030

 

436
00:13:43,030 --> 00:13:46,949

 

437
00:13:46,949 --> 00:13:49,430

 

438
00:13:49,430 --> 00:13:52,310

 

439
00:13:52,310 --> 00:13:55,030

 

440
00:13:55,030 --> 00:13:57,110

 

441
00:13:57,110 --> 00:13:59,110

 

442
00:13:59,110 --> 00:14:01,350

 

443
00:14:01,350 --> 00:14:03,030

 

444
00:14:03,030 --> 00:14:05,829

 

445
00:14:05,829 --> 00:14:07,910

 

446
00:14:07,910 --> 00:14:10,629

 

447
00:14:10,629 --> 00:14:10,639

 

448
00:14:10,639 --> 00:14:15,990


449
00:14:15,990 --> 00:14:16,000

 

450
00:14:16,000 --> 00:14:17,030


451
00:14:17,030 --> 00:14:17,040

 

452
00:14:17,040 --> 00:14:17,490


453
00:14:17,490 --> 00:14:17,500

 

454
00:14:17,500 --> 00:14:19,990


455
00:14:19,990 --> 00:14:23,110

 

456
00:14:23,110 --> 00:14:23,120

 

457
00:14:23,120 --> 00:14:24,150


458
00:14:24,150 --> 00:14:24,160

 

459
00:14:24,160 --> 00:14:24,530


460
00:14:24,530 --> 00:14:24,540

 

461
00:14:24,540 --> 00:14:27,509


462
00:14:27,509 --> 00:14:30,470

 

463
00:14:30,470 --> 00:14:31,750

 

464
00:14:31,750 --> 00:14:33,829

 

465
00:14:33,829 --> 00:14:36,230

 

466
00:14:36,230 --> 00:14:38,949

 

467
00:14:38,949 --> 00:14:41,110

 

468
00:14:41,110 --> 00:14:42,790

 

469
00:14:42,790 --> 00:14:44,710

 

470
00:14:44,710 --> 00:14:47,110

 

471
00:14:47,110 --> 00:14:49,030

 

472
00:14:49,030 --> 00:14:51,189

 

473
00:14:51,189 --> 00:14:52,389

 

474
00:14:52,389 --> 00:14:53,990

 

475
00:14:53,990 --> 00:14:55,829

 

476
00:14:55,829 --> 00:14:57,750

 

477
00:14:57,750 --> 00:14:59,030

 

478
00:14:59,030 --> 00:15:01,189

 

479
00:15:01,189 --> 00:15:03,269

 

480
00:15:03,269 --> 00:15:06,629

 

481
00:15:06,629 --> 00:15:08,550

 

482
00:15:08,550 --> 00:15:10,710

 

483
00:15:10,710 --> 00:15:13,990

 

484
00:15:13,990 --> 00:15:16,389

 

485
00:15:16,389 --> 00:15:17,590

 

486
00:15:17,590 --> 00:15:20,389

 

487
00:15:20,389 --> 00:15:22,069

 

488
00:15:22,069 --> 00:15:25,590

 

489
00:15:25,590 --> 00:15:27,750

 

490
00:15:27,750 --> 00:15:30,790

 

491
00:15:30,790 --> 00:15:33,509

 

492
00:15:33,509 --> 00:15:35,509

 

493
00:15:35,509 --> 00:15:35,519

 

494
00:15:35,519 --> 00:15:36,629


495
00:15:36,629 --> 00:15:38,870

 

496
00:15:38,870 --> 00:15:42,310

 

497
00:15:42,310 --> 00:15:44,790

 

498
00:15:44,790 --> 00:15:47,509

 

499
00:15:47,509 --> 00:15:49,430

 

500
00:15:49,430 --> 00:15:52,230

 

501
00:15:52,230 --> 00:15:54,550

 

502
00:15:54,550 --> 00:15:54,560

 

503
00:15:54,560 --> 00:15:55,829


504
00:15:55,829 --> 00:15:58,470

 

505
00:15:58,470 --> 00:16:00,870

 

506
00:16:00,870 --> 00:16:04,389

 

507
00:16:04,389 --> 00:16:04,399

 

508
00:16:04,399 --> 00:16:05,189


509
00:16:05,189 --> 00:16:07,030

 

510
00:16:07,030 --> 00:16:09,590

 

511
00:16:09,590 --> 00:16:11,910

 

512
00:16:11,910 --> 00:16:11,920

 

513
00:16:11,920 --> 00:16:13,189


514
00:16:13,189 --> 00:16:15,030

 

515
00:16:15,030 --> 00:16:18,629

 

516
00:16:18,629 --> 00:16:20,710

 

517
00:16:20,710 --> 00:16:23,030

 

518
00:16:23,030 --> 00:16:23,040

 

519
00:16:23,040 --> 00:16:24,230


520
00:16:24,230 --> 00:16:25,910

 

521
00:16:25,910 --> 00:16:27,829

 

522
00:16:27,829 --> 00:16:27,839

 

523
00:16:27,839 --> 00:16:28,870


524
00:16:28,870 --> 00:16:30,550

 

525
00:16:30,550 --> 00:16:32,870

 

526
00:16:32,870 --> 00:16:36,230

 

527
00:16:36,230 --> 00:16:38,629

 

528
00:16:38,629 --> 00:16:38,639

 

529
00:16:38,639 --> 00:16:39,590


530
00:16:39,590 --> 00:16:42,150

 

531
00:16:42,150 --> 00:16:45,350

 

532
00:16:45,350 --> 00:16:46,389

 

533
00:16:46,389 --> 00:16:46,399

 

534
00:16:46,399 --> 00:16:48,150


535
00:16:48,150 --> 00:16:51,670

 

536
00:16:51,670 --> 00:16:55,030

 

537
00:16:55,030 --> 00:16:57,030

 

538
00:16:57,030 --> 00:17:00,629

 

539
00:17:00,629 --> 00:17:03,509

 

540
00:17:03,509 --> 00:17:05,829

 

541
00:17:05,829 --> 00:17:08,309

 

542
00:17:08,309 --> 00:17:08,319

 

543
00:17:08,319 --> 00:17:09,350


544
00:17:09,350 --> 00:17:13,270

 

545
00:17:13,270 --> 00:17:15,189

 

546
00:17:15,189 --> 00:17:17,350

 

547
00:17:17,350 --> 00:17:19,909

 

548
00:17:19,909 --> 00:17:24,230

 

549
00:17:24,230 --> 00:17:26,949

 

550
00:17:26,949 --> 00:17:28,470

 

551
00:17:28,470 --> 00:17:32,230

 

552
00:17:32,230 --> 00:17:33,990

 

553
00:17:33,990 --> 00:17:37,029

 

554
00:17:37,029 --> 00:17:37,039

 

555
00:17:37,039 --> 00:17:38,630


556
00:17:38,630 --> 00:17:40,789

 

557
00:17:40,789 --> 00:17:44,310

 

558
00:17:44,310 --> 00:17:46,470

 

559
00:17:46,470 --> 00:17:46,480

 

560
00:17:46,480 --> 00:17:49,110


561
00:17:49,110 --> 00:17:50,789

 

562
00:17:50,789 --> 00:17:53,750

 

563
00:17:53,750 --> 00:17:55,830

 

564
00:17:55,830 --> 00:17:59,270

 

565
00:17:59,270 --> 00:18:01,270

 

566
00:18:01,270 --> 00:18:03,590

 

567
00:18:03,590 --> 00:18:05,029

 

568
00:18:05,029 --> 00:18:08,549

 

569
00:18:08,549 --> 00:18:08,559

 

570
00:18:08,559 --> 00:18:10,710


571
00:18:10,710 --> 00:18:12,950

 

572
00:18:12,950 --> 00:18:14,870

 

573
00:18:14,870 --> 00:18:16,390

 

574
00:18:16,390 --> 00:18:18,310

 

575
00:18:18,310 --> 00:18:22,150

 

576
00:18:22,150 --> 00:18:23,750

 

577
00:18:23,750 --> 00:18:23,760

 

578
00:18:23,760 --> 00:18:24,870


579
00:18:24,870 --> 00:18:24,880

 

580
00:18:24,880 --> 00:18:27,190


581
00:18:27,190 --> 00:18:29,830

 

582
00:18:29,830 --> 00:18:31,669

 

583
00:18:31,669 --> 00:18:32,950

 

584
00:18:32,950 --> 00:18:34,390

 

585
00:18:34,390 --> 00:18:35,830

 

586
00:18:35,830 --> 00:18:38,230

 

587
00:18:38,230 --> 00:18:40,070

 

588
00:18:40,070 --> 00:18:41,350

 

589
00:18:41,350 --> 00:18:43,270

 

590
00:18:43,270 --> 00:18:45,590

 

591
00:18:45,590 --> 00:18:46,789

 

592
00:18:46,789 --> 00:18:49,830

 

593
00:18:49,830 --> 00:18:51,510

 

594
00:18:51,510 --> 00:18:51,520

 

595
00:18:51,520 --> 00:18:52,390


596
00:18:52,390 --> 00:18:52,400

 

597
00:18:52,400 --> 00:18:53,909


598
00:18:53,909 --> 00:18:57,510

 

599
00:18:57,510 --> 00:18:59,909

 

600
00:18:59,909 --> 00:19:01,669

 

601
00:19:01,669 --> 00:19:03,190

 

602
00:19:03,190 --> 00:19:05,029

 

603
00:19:05,029 --> 00:19:05,039

 

604
00:19:05,039 --> 00:19:07,270


605
00:19:07,270 --> 00:19:08,710

 

606
00:19:08,710 --> 00:19:11,510

 

607
00:19:11,510 --> 00:19:14,310

 

608
00:19:14,310 --> 00:19:14,320

 

609
00:19:14,320 --> 00:19:16,070


610
00:19:16,070 --> 00:19:17,990

 

611
00:19:17,990 --> 00:19:20,950

 

612
00:19:20,950 --> 00:19:20,960

 

613
00:19:20,960 --> 00:19:21,320


614
00:19:21,320 --> 00:19:21,330

 

615
00:19:21,330 --> 00:19:23,590


616
00:19:23,590 --> 00:19:25,590

 

617
00:19:25,590 --> 00:19:27,270

 

618
00:19:27,270 --> 00:19:29,350

 

619
00:19:29,350 --> 00:19:30,710

 

620
00:19:30,710 --> 00:19:30,720

 

621
00:19:30,720 --> 00:19:31,990


622
00:19:31,990 --> 00:19:35,590

 

623
00:19:35,590 --> 00:19:37,110

 

624
00:19:37,110 --> 00:19:38,549

 

625
00:19:38,549 --> 00:19:40,390

 

626
00:19:40,390 --> 00:19:42,789

 

627
00:19:42,789 --> 00:19:44,950

 

628
00:19:44,950 --> 00:19:46,870

 

629
00:19:46,870 --> 00:19:46,880

 

630
00:19:46,880 --> 00:19:48,789


631
00:19:48,789 --> 00:19:51,270

 

632
00:19:51,270 --> 00:19:53,270

 

633
00:19:53,270 --> 00:19:54,870

 

634
00:19:54,870 --> 00:19:57,510

 

635
00:19:57,510 --> 00:19:57,520

 

636
00:19:57,520 --> 00:19:59,190


637
00:19:59,190 --> 00:20:01,830

 

638
00:20:01,830 --> 00:20:04,470

 

639
00:20:04,470 --> 00:20:07,590

 

640
00:20:07,590 --> 00:20:07,600

 

641
00:20:07,600 --> 00:20:08,470


642
00:20:08,470 --> 00:20:11,830

 

643
00:20:11,830 --> 00:20:13,510

 

644
00:20:13,510 --> 00:20:15,270

 

645
00:20:15,270 --> 00:20:16,789

 

646
00:20:16,789 --> 00:20:18,470

 

647
00:20:18,470 --> 00:20:20,390

 

648
00:20:20,390 --> 00:20:21,750

 

649
00:20:21,750 --> 00:20:23,110

 

650
00:20:23,110 --> 00:20:25,510

 

651
00:20:25,510 --> 00:20:27,110

 

652
00:20:27,110 --> 00:20:29,909

 

653
00:20:29,909 --> 00:20:29,919

 

654
00:20:29,919 --> 00:20:32,070


655
00:20:32,070 --> 00:20:34,710

 

656
00:20:34,710 --> 00:20:36,310

 

657
00:20:36,310 --> 00:20:38,710

 

658
00:20:38,710 --> 00:20:41,270

 

659
00:20:41,270 --> 00:20:42,390

 

660
00:20:42,390 --> 00:20:42,400

 

661
00:20:42,400 --> 00:20:43,590


662
00:20:43,590 --> 00:20:46,149

 

663
00:20:46,149 --> 00:20:49,190

 

664
00:20:49,190 --> 00:20:50,630

 

665
00:20:50,630 --> 00:20:53,830

 

666
00:20:53,830 --> 00:20:55,990

 

667
00:20:55,990 --> 00:20:57,350

 

668
00:20:57,350 --> 00:20:59,190

 

669
00:20:59,190 --> 00:21:01,190

 

670
00:21:01,190 --> 00:21:02,630

 

671
00:21:02,630 --> 00:21:04,630

 

672
00:21:04,630 --> 00:21:05,990

 

673
00:21:05,990 --> 00:21:08,310

 

674
00:21:08,310 --> 00:21:09,909

 

675
00:21:09,909 --> 00:21:13,190

 

676
00:21:13,190 --> 00:21:15,029

 

677
00:21:15,029 --> 00:21:17,430

 

678
00:21:17,430 --> 00:21:18,789

 

679
00:21:18,789 --> 00:21:21,510

 

680
00:21:21,510 --> 00:21:24,630

 

681
00:21:24,630 --> 00:21:26,149

 

682
00:21:26,149 --> 00:21:28,149

 

683
00:21:28,149 --> 00:21:29,909

 

684
00:21:29,909 --> 00:21:31,350

 

685
00:21:31,350 --> 00:21:31,360

 

686
00:21:31,360 --> 00:21:32,230


687
00:21:32,230 --> 00:21:34,230

 

688
00:21:34,230 --> 00:21:37,669

 

689
00:21:37,669 --> 00:21:39,909

 

690
00:21:39,909 --> 00:21:42,230

 

691
00:21:42,230 --> 00:21:45,590

 

692
00:21:45,590 --> 00:21:50,149

 

693
00:21:50,149 --> 00:21:51,830

 

694
00:21:51,830 --> 00:21:53,750

 

695
00:21:53,750 --> 00:21:55,190

 

696
00:21:55,190 --> 00:21:56,710

 

697
00:21:56,710 --> 00:21:58,789

 

698
00:21:58,789 --> 00:22:01,990

 

699
00:22:01,990 --> 00:22:02,000

 

700
00:22:02,000 --> 00:22:02,870


701
00:22:02,870 --> 00:22:04,470

 

702
00:22:04,470 --> 00:22:06,310

 

703
00:22:06,310 --> 00:22:08,710

 

704
00:22:08,710 --> 00:22:11,830

 

705
00:22:11,830 --> 00:22:13,190

 

706
00:22:13,190 --> 00:22:15,430

 

707
00:22:15,430 --> 00:22:15,440

 

708
00:22:15,440 --> 00:22:20,630


709
00:22:20,630 --> 00:22:24,630

 

710
00:22:24,630 --> 00:22:27,350

 

711
00:22:27,350 --> 00:22:29,350

 

712
00:22:29,350 --> 00:22:32,710

 

713
00:22:32,710 --> 00:22:34,149

 

714
00:22:34,149 --> 00:22:35,830

 

715
00:22:35,830 --> 00:22:38,390

 

716
00:22:38,390 --> 00:22:41,029

 

717
00:22:41,029 --> 00:22:43,909

 

718
00:22:43,909 --> 00:22:43,919

 

719
00:22:43,919 --> 00:22:44,340


720
00:22:44,340 --> 00:22:44,350

 

721
00:22:44,350 --> 00:22:46,390


722
00:22:46,390 --> 00:22:50,230

 

723
00:22:50,230 --> 00:22:51,830

 

724
00:22:51,830 --> 00:22:53,350

 

725
00:22:53,350 --> 00:22:55,830

 

726
00:22:55,830 --> 00:22:57,909

 

727
00:22:57,909 --> 00:23:00,149

 

728
00:23:00,149 --> 00:23:00,159

 

729
00:23:00,159 --> 00:23:01,110


730
00:23:01,110 --> 00:23:01,120

 

731
00:23:01,120 --> 00:23:02,070


732
00:23:02,070 --> 00:23:04,870

 

733
00:23:04,870 --> 00:23:04,880

 

734
00:23:04,880 --> 00:23:07,190


735
00:23:07,190 --> 00:23:10,549

 

736
00:23:10,549 --> 00:23:11,909

 

737
00:23:11,909 --> 00:23:13,990

 

738
00:23:13,990 --> 00:23:15,029

 

739
00:23:15,029 --> 00:23:15,039

 

740
00:23:15,039 --> 00:23:16,470


741
00:23:16,470 --> 00:23:16,480

 

742
00:23:16,480 --> 00:23:17,590


743
00:23:17,590 --> 00:23:21,029

 

744
00:23:21,029 --> 00:23:24,070

 

745
00:23:24,070 --> 00:23:26,070

 

746
00:23:26,070 --> 00:23:27,270

 

747
00:23:27,270 --> 00:23:27,280

 

748
00:23:27,280 --> 00:23:28,310


749
00:23:28,310 --> 00:23:30,950

 

750
00:23:30,950 --> 00:23:32,789

 

751
00:23:32,789 --> 00:23:34,549

 

752
00:23:34,549 --> 00:23:36,230

 

753
00:23:36,230 --> 00:23:36,240

 

754
00:23:36,240 --> 00:23:37,909


755
00:23:37,909 --> 00:23:37,919

 

756
00:23:37,919 --> 00:23:39,669


757
00:23:39,669 --> 00:23:39,679

 

758
00:23:39,679 --> 00:23:40,710


759
00:23:40,710 --> 00:23:42,310

 

760
00:23:42,310 --> 00:23:43,990

 

761
00:23:43,990 --> 00:23:45,750

 

762
00:23:45,750 --> 00:23:47,590

 

763
00:23:47,590 --> 00:23:50,310

 

764
00:23:50,310 --> 00:23:52,149

 

765
00:23:52,149 --> 00:23:52,159

 

766
00:23:52,159 --> 00:23:53,750


767
00:23:53,750 --> 00:23:56,310

 

768
00:23:56,310 --> 00:23:58,710

 

769
00:23:58,710 --> 00:24:01,510

 

770
00:24:01,510 --> 00:24:03,510

 

771
00:24:03,510 --> 00:24:05,350

 

772
00:24:05,350 --> 00:24:06,789

 

773
00:24:06,789 --> 00:24:09,430

 

774
00:24:09,430 --> 00:24:11,510

 

775
00:24:11,510 --> 00:24:13,750

 

776
00:24:13,750 --> 00:24:15,830

 

777
00:24:15,830 --> 00:24:18,070

 

778
00:24:18,070 --> 00:24:20,710

 

779
00:24:20,710 --> 00:24:22,310

 

780
00:24:22,310 --> 00:24:23,990

 

781
00:24:23,990 --> 00:24:25,750

 

782
00:24:25,750 --> 00:24:28,950

 

783
00:24:28,950 --> 00:24:31,269

 

784
00:24:31,269 --> 00:24:33,750

 

785
00:24:33,750 --> 00:24:35,510

 

786
00:24:35,510 --> 00:24:39,269

 

787
00:24:39,269 --> 00:24:41,830

 

788
00:24:41,830 --> 00:24:43,830

 

789
00:24:43,830 --> 00:24:43,840

 

790
00:24:43,840 --> 00:24:45,269


791
00:24:45,269 --> 00:24:48,070

 

792
00:24:48,070 --> 00:24:50,950

 

793
00:24:50,950 --> 00:24:52,470

 

794
00:24:52,470 --> 00:24:54,070

 

795
00:24:54,070 --> 00:24:54,080

 

796
00:24:54,080 --> 00:24:55,350


797
00:24:55,350 --> 00:24:57,029

 

798
00:24:57,029 --> 00:25:00,549

 

799
00:25:00,549 --> 00:25:03,430

 

800
00:25:03,430 --> 00:25:05,430

 

801
00:25:05,430 --> 00:25:06,710

 

802
00:25:06,710 --> 00:25:08,789

 

803
00:25:08,789 --> 00:25:10,630

 

804
00:25:10,630 --> 00:25:15,750

 

805
00:25:15,750 --> 00:25:17,590

 

806
00:25:17,590 --> 00:25:19,669

 

807
00:25:19,669 --> 00:25:21,430

 

808
00:25:21,430 --> 00:25:23,669

 

809
00:25:23,669 --> 00:25:25,029

 

810
00:25:25,029 --> 00:25:26,470

 

811
00:25:26,470 --> 00:25:28,950

 

812
00:25:28,950 --> 00:25:31,750

 

813
00:25:31,750 --> 00:25:34,149

 

814
00:25:34,149 --> 00:25:37,269

 

815
00:25:37,269 --> 00:25:37,279

 

816
00:25:37,279 --> 00:25:38,310


817
00:25:38,310 --> 00:25:40,390

 

818
00:25:40,390 --> 00:25:42,950

 

819
00:25:42,950 --> 00:25:45,269

 

820
00:25:45,269 --> 00:25:47,269

 

821
00:25:47,269 --> 00:25:49,110

 

822
00:25:49,110 --> 00:25:51,110

 

823
00:25:51,110 --> 00:25:52,470

 

824
00:25:52,470 --> 00:25:55,510

 

825
00:25:55,510 --> 00:25:57,269

 

826
00:25:57,269 --> 00:26:00,230

 

827
00:26:00,230 --> 00:26:04,230

 

828
00:26:04,230 --> 00:26:04,240

 

829
00:26:04,240 --> 00:26:06,149


830
00:26:06,149 --> 00:26:07,990

 

831
00:26:07,990 --> 00:26:11,350

 

832
00:26:11,350 --> 00:26:13,029

 

833
00:26:13,029 --> 00:26:14,230

 

834
00:26:14,230 --> 00:26:16,470

 

835
00:26:16,470 --> 00:26:17,830

 

836
00:26:17,830 --> 00:26:19,430

 

837
00:26:19,430 --> 00:26:21,909

 

838
00:26:21,909 --> 00:26:23,830

 

839
00:26:23,830 --> 00:26:25,909

 

840
00:26:25,909 --> 00:26:27,830

 

841
00:26:27,830 --> 00:26:29,990

 

842
00:26:29,990 --> 00:26:31,510

 

843
00:26:31,510 --> 00:26:33,190

 

844
00:26:33,190 --> 00:26:34,310

 

845
00:26:34,310 --> 00:26:37,909

 

846
00:26:37,909 --> 00:26:39,990

 

847
00:26:39,990 --> 00:26:41,350

 

848
00:26:41,350 --> 00:26:42,789

 

849
00:26:42,789 --> 00:26:42,799

 

850
00:26:42,799 --> 00:26:43,909


851
00:26:43,909 --> 00:26:46,549

 

852
00:26:46,549 --> 00:26:46,559

 

853
00:26:46,559 --> 00:26:48,149


854
00:26:48,149 --> 00:26:51,669

 

855
00:26:51,669 --> 00:26:53,590

 

856
00:26:53,590 --> 00:26:53,600

 

857
00:26:53,600 --> 00:26:54,549


858
00:26:54,549 --> 00:26:57,990

 

859
00:26:57,990 --> 00:27:00,230

 

860
00:27:00,230 --> 00:27:01,510

 

861
00:27:01,510 --> 00:27:04,549

 

862
00:27:04,549 --> 00:27:07,029

 

863
00:27:07,029 --> 00:27:09,590

 

864
00:27:09,590 --> 00:27:11,029

 

865
00:27:11,029 --> 00:27:13,029

 

866
00:27:13,029 --> 00:27:15,029

 

867
00:27:15,029 --> 00:27:17,110

 

868
00:27:17,110 --> 00:27:19,190

 

869
00:27:19,190 --> 00:27:20,710

 

870
00:27:20,710 --> 00:27:22,149

 

871
00:27:22,149 --> 00:27:24,149

 

872
00:27:24,149 --> 00:27:25,990

 

873
00:27:25,990 --> 00:27:27,590

 

874
00:27:27,590 --> 00:27:29,269

 

875
00:27:29,269 --> 00:27:31,430

 

876
00:27:31,430 --> 00:27:35,110

 

877
00:27:35,110 --> 00:27:37,669

 

878
00:27:37,669 --> 00:27:41,830

 

879
00:27:41,830 --> 00:27:44,470

 

880
00:27:44,470 --> 00:27:47,430

 

881
00:27:47,430 --> 00:27:49,990

 

882
00:27:49,990 --> 00:27:52,389

 

883
00:27:52,389 --> 00:27:54,710

 

884
00:27:54,710 --> 00:27:57,590

 

885
00:27:57,590 --> 00:27:59,269

 

886
00:27:59,269 --> 00:28:00,389

 

887
00:28:00,389 --> 00:28:00,399

 

888
00:28:00,399 --> 00:28:01,430


889
00:28:01,430 --> 00:28:03,029

 

890
00:28:03,029 --> 00:28:05,269

 

891
00:28:05,269 --> 00:28:07,990

 

892
00:28:07,990 --> 00:28:09,510

 

893
00:28:09,510 --> 00:28:11,590

 

894
00:28:11,590 --> 00:28:12,950

 

895
00:28:12,950 --> 00:28:14,389

 

896
00:28:14,389 --> 00:28:14,399

 

897
00:28:14,399 --> 00:28:16,149


898
00:28:16,149 --> 00:28:19,430

 

899
00:28:19,430 --> 00:28:20,789

 

900
00:28:20,789 --> 00:28:22,870

 

901
00:28:22,870 --> 00:28:22,880

 

902
00:28:22,880 --> 00:28:24,149


903
00:28:24,149 --> 00:28:25,110

 

904
00:28:25,110 --> 00:28:26,870

 

905
00:28:26,870 --> 00:28:28,630

 

906
00:28:28,630 --> 00:28:31,190

 

907
00:28:31,190 --> 00:28:32,710

 

908
00:28:32,710 --> 00:28:34,310

 

909
00:28:34,310 --> 00:28:36,470

 

910
00:28:36,470 --> 00:28:36,480

 

911
00:28:36,480 --> 00:28:38,549


912
00:28:38,549 --> 00:28:40,630

 

913
00:28:40,630 --> 00:28:44,870

 

914
00:28:44,870 --> 00:28:47,830

 

915
00:28:47,830 --> 00:28:49,269

 

916
00:28:49,269 --> 00:28:51,909

 

917
00:28:51,909 --> 00:28:53,750

 

918
00:28:53,750 --> 00:28:57,750

 

919
00:28:57,750 --> 00:28:59,510

 

920
00:28:59,510 --> 00:29:01,269

 

921
00:29:01,269 --> 00:29:02,870

 

922
00:29:02,870 --> 00:29:02,880

 

923
00:29:02,880 --> 00:29:04,389


924
00:29:04,389 --> 00:29:06,950

 

925
00:29:06,950 --> 00:29:09,990

 

926
00:29:09,990 --> 00:29:11,430

 

927
00:29:11,430 --> 00:29:12,630

 

928
00:29:12,630 --> 00:29:16,870

 

929
00:29:16,870 --> 00:29:19,350

 

930
00:29:19,350 --> 00:29:19,360

 

931
00:29:19,360 --> 00:29:20,630


932
00:29:20,630 --> 00:29:22,549

 

933
00:29:22,549 --> 00:29:24,950

 

934
00:29:24,950 --> 00:29:26,870

 

935
00:29:26,870 --> 00:29:31,430

 

936
00:29:31,430 --> 00:29:33,110

 

937
00:29:33,110 --> 00:29:35,590

 

938
00:29:35,590 --> 00:29:38,149

 

939
00:29:38,149 --> 00:29:39,750

 

940
00:29:39,750 --> 00:29:41,190

 

941
00:29:41,190 --> 00:29:43,190

 

942
00:29:43,190 --> 00:29:45,909

 

943
00:29:45,909 --> 00:29:49,430

 

944
00:29:49,430 --> 00:29:50,950

 

945
00:29:50,950 --> 00:29:52,549

 

946
00:29:52,549 --> 00:29:52,559

 

947
00:29:52,559 --> 00:29:53,430


948
00:29:53,430 --> 00:29:55,190

 

949
00:29:55,190 --> 00:29:57,110

 

950
00:29:57,110 --> 00:29:58,710

 

951
00:29:58,710 --> 00:29:58,720

 

952
00:29:58,720 --> 00:30:00,149


953
00:30:00,149 --> 00:30:04,389

 

954
00:30:04,389 --> 00:30:04,399

 

955
00:30:04,399 --> 00:30:06,710


956
00:30:06,710 --> 00:30:08,870

 

957
00:30:08,870 --> 00:30:10,870

 

958
00:30:10,870 --> 00:30:12,950

 

959
00:30:12,950 --> 00:30:16,950

 

960
00:30:16,950 --> 00:30:19,269

 

961
00:30:19,269 --> 00:30:22,870

 

962
00:30:22,870 --> 00:30:22,880

 

963
00:30:22,880 --> 00:30:24,630


964
00:30:24,630 --> 00:30:27,029

 

965
00:30:27,029 --> 00:30:29,190

 

966
00:30:29,190 --> 00:30:31,510

 

967
00:30:31,510 --> 00:30:32,630

 

968
00:30:32,630 --> 00:30:34,870

 

969
00:30:34,870 --> 00:30:36,710

 

970
00:30:36,710 --> 00:30:39,750

 

971
00:30:39,750 --> 00:30:41,190

 

972
00:30:41,190 --> 00:30:43,110

 

973
00:30:43,110 --> 00:30:45,669

 

974
00:30:45,669 --> 00:30:47,590

 

975
00:30:47,590 --> 00:30:52,070

 

976
00:30:52,070 --> 00:30:52,080

 

977
00:30:52,080 --> 00:30:53,110


978
00:30:53,110 --> 00:30:56,549

 

979
00:30:56,549 --> 00:30:58,389

 

980
00:30:58,389 --> 00:31:01,029

 

981
00:31:01,029 --> 00:31:03,029

 

982
00:31:03,029 --> 00:31:06,549

 

983
00:31:06,549 --> 00:31:09,669

 

984
00:31:09,669 --> 00:31:13,269

 

985
00:31:13,269 --> 00:31:15,590

 

986
00:31:15,590 --> 00:31:18,789

 

987
00:31:18,789 --> 00:31:18,799

 

988
00:31:18,799 --> 00:31:21,990


989
00:31:21,990 --> 00:31:23,830

 

990
00:31:23,830 --> 00:31:25,110

 

991
00:31:25,110 --> 00:31:27,430

 

992
00:31:27,430 --> 00:31:30,789

 

993
00:31:30,789 --> 00:31:32,389

 

994
00:31:32,389 --> 00:31:35,350

 

995
00:31:35,350 --> 00:31:38,549

 

996
00:31:38,549 --> 00:31:40,549

 

997
00:31:40,549 --> 00:31:42,230

 

998
00:31:42,230 --> 00:31:44,149

 

999
00:31:44,149 --> 00:31:46,870

 

1000
00:31:46,870 --> 00:31:46,880

 

1001
00:31:46,880 --> 00:31:47,909


1002
00:31:47,909 --> 00:31:50,070

 

1003
00:31:50,070 --> 00:31:53,110

 

1004
00:31:53,110 --> 00:31:55,110

 

1005
00:31:55,110 --> 00:31:57,509

 

1006
00:31:57,509 --> 00:31:59,590

 

1007
00:31:59,590 --> 00:31:59,600

 

1008
00:31:59,600 --> 00:32:01,269


1009
00:32:01,269 --> 00:32:04,070

 

1010
00:32:04,070 --> 00:32:05,269

 

1011
00:32:05,269 --> 00:32:05,279

 

1012
00:32:05,279 --> 00:32:07,029


1013
00:32:07,029 --> 00:32:07,039

 

1014
00:32:07,039 --> 00:32:08,549


1015
00:32:08,549 --> 00:32:08,559

 

1016
00:32:08,559 --> 00:32:09,669


1017
00:32:09,669 --> 00:32:11,990

 

1018
00:32:11,990 --> 00:32:15,669

 

1019
00:32:15,669 --> 00:32:17,430

 

1020
00:32:17,430 --> 00:32:17,440

 

1021
00:32:17,440 --> 00:32:18,630


1022
00:32:18,630 --> 00:32:20,630

 

1023
00:32:20,630 --> 00:32:25,430

 

1024
00:32:25,430 --> 00:32:27,190

 

1025
00:32:27,190 --> 00:32:30,070

 

1026
00:32:30,070 --> 00:32:32,070

 

1027
00:32:32,070 --> 00:32:35,190

 

1028
00:32:35,190 --> 00:32:37,029

 

1029
00:32:37,029 --> 00:32:38,870

 

1030
00:32:38,870 --> 00:32:42,230

 

1031
00:32:42,230 --> 00:32:45,830

 

1032
00:32:45,830 --> 00:32:47,350

 

1033
00:32:47,350 --> 00:32:49,750

 

1034
00:32:49,750 --> 00:32:52,070

 

1035
00:32:52,070 --> 00:32:53,909

 

1036
00:32:53,909 --> 00:32:56,549

 

1037
00:32:56,549 --> 00:33:00,549

 

1038
00:33:00,549 --> 00:33:00,559

 

1039
00:33:00,559 --> 00:33:03,669


1040
00:33:03,669 --> 00:33:05,190

 

1041
00:33:05,190 --> 00:33:06,870

 

1042
00:33:06,870 --> 00:33:06,880

 

1043
00:33:06,880 --> 00:33:07,990


1044
00:33:07,990 --> 00:33:11,269

 

1045
00:33:11,269 --> 00:33:13,110

 

1046
00:33:13,110 --> 00:33:15,190

 

1047
00:33:15,190 --> 00:33:16,470

 

1048
00:33:16,470 --> 00:33:18,789

 

1049
00:33:18,789 --> 00:33:20,470

 

1050
00:33:20,470 --> 00:33:28,630

 

1051
00:33:28,630 --> 00:33:30,870

 

1052
00:33:30,870 --> 00:33:34,149

 

1053
00:33:34,149 --> 00:33:34,159

 

1054
00:33:34,159 --> 00:33:36,830


1055
00:33:36,830 --> 00:33:36,840

 

1056
00:33:36,840 --> 00:33:38,710


1057
00:33:38,710 --> 00:33:40,070

 

1058
00:33:40,070 --> 00:33:42,549

 

1059
00:33:42,549 --> 00:33:44,789

 

1060
00:33:44,789 --> 00:33:47,190

 

1061
00:33:47,190 --> 00:33:49,509

 

1062
00:33:49,509 --> 00:33:52,070

 

1063
00:33:52,070 --> 00:33:53,909

 

1064
00:33:53,909 --> 00:33:56,630

 

1065
00:33:56,630 --> 00:33:58,310

 

1066
00:33:58,310 --> 00:33:58,320

 

1067
00:33:58,320 --> 00:33:59,350


1068
00:33:59,350 --> 00:33:59,360

 

1069
00:33:59,360 --> 00:34:03,990


1070
00:34:03,990 --> 00:34:05,590

 

1071
00:34:05,590 --> 00:34:08,710

 

1072
00:34:08,710 --> 00:34:11,829

 

1073
00:34:11,829 --> 00:34:13,270

 

1074
00:34:13,270 --> 00:34:14,869

 

1075
00:34:14,869 --> 00:34:17,750

 

1076
00:34:17,750 --> 00:34:21,750

 

1077
00:34:21,750 --> 00:34:23,510

 

1078
00:34:23,510 --> 00:34:26,310

 

1079
00:34:26,310 --> 00:34:27,589

 

1080
00:34:27,589 --> 00:34:29,030

 

1081
00:34:29,030 --> 00:34:31,030

 

1082
00:34:31,030 --> 00:34:31,909

 

1083
00:34:31,909 --> 00:34:34,950

 

1084
00:34:34,950 --> 00:34:36,310

 

1085
00:34:36,310 --> 00:34:38,470

 

1086
00:34:38,470 --> 00:34:40,790

 

1087
00:34:40,790 --> 00:34:42,710

 

1088
00:34:42,710 --> 00:34:44,230

 

1089
00:34:44,230 --> 00:34:45,430

 

1090
00:34:45,430 --> 00:34:46,550

 

1091
00:34:46,550 --> 00:34:48,230

 

1092
00:34:48,230 --> 00:34:49,829

 

1093
00:34:49,829 --> 00:34:51,669

 

1094
00:34:51,669 --> 00:34:53,190

 

1095
00:34:53,190 --> 00:34:54,790

 

1096
00:34:54,790 --> 00:34:57,349

 

1097
00:34:57,349 --> 00:34:58,550

 

1098
00:34:58,550 --> 00:34:58,560

 

1099
00:34:58,560 --> 00:34:59,990


1100
00:34:59,990 --> 00:35:01,589

 

1101
00:35:01,589 --> 00:35:03,270

 

1102
00:35:03,270 --> 00:35:07,510

 

1103
00:35:07,510 --> 00:35:07,520

 

1104
00:35:07,520 --> 00:35:08,550


1105
00:35:08,550 --> 00:35:10,390

 

1106
00:35:10,390 --> 00:35:10,400

 

1107
00:35:10,400 --> 00:35:13,109


1108
00:35:13,109 --> 00:35:13,119

 

1109
00:35:13,119 --> 00:35:14,390


1110
00:35:14,390 --> 00:35:19,990

 

1111
00:35:19,990 --> 00:35:24,310

 

1112
00:35:24,310 --> 00:35:26,870

 

1113
00:35:26,870 --> 00:35:29,270

 

1114
00:35:29,270 --> 00:35:31,589

 

1115
00:35:31,589 --> 00:35:33,589

 

1116
00:35:33,589 --> 00:35:35,670

 

1117
00:35:35,670 --> 00:35:39,910

 

1118
00:35:39,910 --> 00:35:41,990

 

1119
00:35:41,990 --> 00:35:49,510

 

1120
00:35:49,510 --> 00:35:51,750

 

1121
00:35:51,750 --> 00:35:55,349

 

1122
00:35:55,349 --> 00:35:56,710

 

1123
00:35:56,710 --> 00:35:56,720

 

1124
00:35:56,720 --> 00:35:58,710


1125
00:35:58,710 --> 00:36:00,470

 

1126
00:36:00,470 --> 00:36:02,230

 

1127
00:36:02,230 --> 00:36:05,190

 

1128
00:36:05,190 --> 00:36:09,510

 

1129
00:36:09,510 --> 00:36:09,520

 

1130
00:36:09,520 --> 00:36:10,470


1131
00:36:10,470 --> 00:36:11,670

 

1132
00:36:11,670 --> 00:36:15,349

 

1133
00:36:15,349 --> 00:36:17,430

 

1134
00:36:17,430 --> 00:36:18,790

 

1135
00:36:18,790 --> 00:36:20,790

 

1136
00:36:20,790 --> 00:36:22,230

 

1137
00:36:22,230 --> 00:36:24,150

 

1138
00:36:24,150 --> 00:36:26,230

 

1139
00:36:26,230 --> 00:36:26,240

 

1140
00:36:26,240 --> 00:36:27,109


1141
00:36:27,109 --> 00:36:28,470

 

1142
00:36:28,470 --> 00:36:30,950

 

1143
00:36:30,950 --> 00:36:33,510

 

1144
00:36:33,510 --> 00:36:37,109

 

1145
00:36:37,109 --> 00:36:38,710

 

1146
00:36:38,710 --> 00:36:40,550

 

1147
00:36:40,550 --> 00:36:40,560

 

1148
00:36:40,560 --> 00:36:41,589


1149
00:36:41,589 --> 00:36:43,910

 

1150
00:36:43,910 --> 00:36:43,920

 

1151
00:36:43,920 --> 00:36:45,430


1152
00:36:45,430 --> 00:36:46,390

 

1153
00:36:46,390 --> 00:36:48,470

 

1154
00:36:48,470 --> 00:36:50,310

 

1155
00:36:50,310 --> 00:36:54,870

 

1156
00:36:54,870 --> 00:36:56,550

 

1157
00:36:56,550 --> 00:36:58,550

 

1158
00:36:58,550 --> 00:37:00,550

 

1159
00:37:00,550 --> 00:37:03,030

 

1160
00:37:03,030 --> 00:37:05,750

 

1161
00:37:05,750 --> 00:37:07,430

 

1162
00:37:07,430 --> 00:37:09,510

 

1163
00:37:09,510 --> 00:37:11,270

 

1164
00:37:11,270 --> 00:37:13,190

 

1165
00:37:13,190 --> 00:37:14,550

 

1166
00:37:14,550 --> 00:37:16,630

 

1167
00:37:16,630 --> 00:37:18,150

 

1168
00:37:18,150 --> 00:37:20,950

 

1169
00:37:20,950 --> 00:37:23,349

 

1170
00:37:23,349 --> 00:37:25,510

 

1171
00:37:25,510 --> 00:37:25,520

 

1172
00:37:25,520 --> 00:37:26,790


1173
00:37:26,790 --> 00:37:26,800

 

1174
00:37:26,800 --> 00:37:27,750


1175
00:37:27,750 --> 00:37:30,069

 

1176
00:37:30,069 --> 00:37:31,670

 

1177
00:37:31,670 --> 00:37:33,910

 

1178
00:37:33,910 --> 00:37:36,870

 

1179
00:37:36,870 --> 00:37:39,030

 

1180
00:37:39,030 --> 00:37:40,470

 

1181
00:37:40,470 --> 00:37:41,750

 

1182
00:37:41,750 --> 00:37:44,710

 

1183
00:37:44,710 --> 00:37:46,150

 

1184
00:37:46,150 --> 00:37:46,160

 

1185
00:37:46,160 --> 00:37:48,230


1186
00:37:48,230 --> 00:37:51,829

 

1187
00:37:51,829 --> 00:37:53,510

 

1188
00:37:53,510 --> 00:37:55,829

 

1189
00:37:55,829 --> 00:37:58,390

 

1190
00:37:58,390 --> 00:38:00,230

 

1191
00:38:00,230 --> 00:38:02,710

 

1192
00:38:02,710 --> 00:38:05,510

 

1193
00:38:05,510 --> 00:38:09,589

 

1194
00:38:09,589 --> 00:38:09,599

 

1195
00:38:09,599 --> 00:38:10,790


1196
00:38:10,790 --> 00:38:13,349

 

1197
00:38:13,349 --> 00:38:13,359

 

1198
00:38:13,359 --> 00:38:14,390


1199
00:38:14,390 --> 00:38:16,310

 

1200
00:38:16,310 --> 00:38:16,320

 

1201
00:38:16,320 --> 00:38:19,910


1202
00:38:19,910 --> 00:38:21,670

 

1203
00:38:21,670 --> 00:38:23,670

 

1204
00:38:23,670 --> 00:38:25,030

 

1205
00:38:25,030 --> 00:38:29,750

 

1206
00:38:29,750 --> 00:38:31,670

 

1207
00:38:31,670 --> 00:38:33,910

 

1208
00:38:33,910 --> 00:38:36,310

 

1209
00:38:36,310 --> 00:38:38,310

 

1210
00:38:38,310 --> 00:38:41,190

 

1211
00:38:41,190 --> 00:38:43,510

 

1212
00:38:43,510 --> 00:38:44,829

 

1213
00:38:44,829 --> 00:38:46,870

 

1214
00:38:46,870 --> 00:38:50,870

 

1215
00:38:50,870 --> 00:38:54,069

 

1216
00:38:54,069 --> 00:38:55,430

 

1217
00:38:55,430 --> 00:38:55,440

 

1218
00:38:55,440 --> 00:38:58,950


1219
00:38:58,950 --> 00:39:00,870

 

1220
00:39:00,870 --> 00:39:00,880

 

1221
00:39:00,880 --> 00:39:02,870


1222
00:39:02,870 --> 00:39:07,430

 

1223
00:39:07,430 --> 00:39:08,870

 

1224
00:39:08,870 --> 00:39:10,150

 

1225
00:39:10,150 --> 00:39:12,550

 

1226
00:39:12,550 --> 00:39:14,710

 

1227
00:39:14,710 --> 00:39:17,270

 

1228
00:39:17,270 --> 00:39:19,829

 

1229
00:39:19,829 --> 00:39:22,470

 

1230
00:39:22,470 --> 00:39:24,470

 

1231
00:39:24,470 --> 00:39:26,230

 

1232
00:39:26,230 --> 00:39:26,240

 

1233
00:39:26,240 --> 00:39:29,030


1234
00:39:29,030 --> 00:39:31,430

 

1235
00:39:31,430 --> 00:39:34,870

 

1236
00:39:34,870 --> 00:39:36,950

 

1237
00:39:36,950 --> 00:39:39,349

 

1238
00:39:39,349 --> 00:39:41,270

 

1239
00:39:41,270 --> 00:39:43,670

 

1240
00:39:43,670 --> 00:39:43,680

 

1241
00:39:43,680 --> 00:39:44,870


1242
00:39:44,870 --> 00:39:47,589

 

1243
00:39:47,589 --> 00:39:49,190

 

1244
00:39:49,190 --> 00:39:49,200

 

1245
00:39:49,200 --> 00:39:50,069


1246
00:39:50,069 --> 00:39:53,190

 

1247
00:39:53,190 --> 00:39:57,190

 

1248
00:39:57,190 --> 00:39:58,870

 

1249
00:39:58,870 --> 00:40:01,270

 

1250
00:40:01,270 --> 00:40:01,280

 

1251
00:40:01,280 --> 00:40:04,150


1252
00:40:04,150 --> 00:40:04,160

 

1253
00:40:04,160 --> 00:40:06,870


1254
00:40:06,870 --> 00:40:06,880

 

1255
00:40:06,880 --> 00:40:07,750


1256
00:40:07,750 --> 00:40:11,270

 

1257
00:40:11,270 --> 00:40:11,280

 

1258
00:40:11,280 --> 00:40:13,270


1259
00:40:13,270 --> 00:40:13,280

 

1260
00:40:13,280 --> 00:40:14,069


1261
00:40:14,069 --> 00:40:16,710

 

1262
00:40:16,710 --> 00:40:19,589

 

1263
00:40:19,589 --> 00:40:19,599

 

1264
00:40:19,599 --> 00:40:20,550


1265
00:40:20,550 --> 00:40:24,710

 

1266
00:40:24,710 --> 00:40:26,630

 

1267
00:40:26,630 --> 00:40:29,109

 

1268
00:40:29,109 --> 00:40:31,270

 

1269
00:40:31,270 --> 00:40:33,670

 

1270
00:40:33,670 --> 00:40:36,390

 

1271
00:40:36,390 --> 00:40:38,390

 

1272
00:40:38,390 --> 00:40:40,550

 

1273
00:40:40,550 --> 00:40:43,430

 

1274
00:40:43,430 --> 00:40:45,190

 

1275
00:40:45,190 --> 00:40:47,190

 

1276
00:40:47,190 --> 00:40:48,870

 

1277
00:40:48,870 --> 00:40:52,309

 

1278
00:40:52,309 --> 00:40:52,319

 

1279
00:40:52,319 --> 00:40:53,270


1280
00:40:53,270 --> 00:40:55,510

 

1281
00:40:55,510 --> 00:40:57,990

 

1282
00:40:57,990 --> 00:41:00,230

 

1283
00:41:00,230 --> 00:41:01,510

 

1284
00:41:01,510 --> 00:41:03,670

 

1285
00:41:03,670 --> 00:41:05,190

 

1286
00:41:05,190 --> 00:41:07,349

 

1287
00:41:07,349 --> 00:41:09,270

 

1288
00:41:09,270 --> 00:41:12,710

 

1289
00:41:12,710 --> 00:41:15,109

 

1290
00:41:15,109 --> 00:41:17,270

 

1291
00:41:17,270 --> 00:41:19,430

 

1292
00:41:19,430 --> 00:41:22,230

 

1293
00:41:22,230 --> 00:41:23,670

 

1294
00:41:23,670 --> 00:41:25,349

 

1295
00:41:25,349 --> 00:41:27,910

 

1296
00:41:27,910 --> 00:41:29,829

 

1297
00:41:29,829 --> 00:41:31,829

 

1298
00:41:31,829 --> 00:41:33,349

 

1299
00:41:33,349 --> 00:41:35,829

 

1300
00:41:35,829 --> 00:41:38,069

 

1301
00:41:38,069 --> 00:41:40,150

 

1302
00:41:40,150 --> 00:41:40,160

 

1303
00:41:40,160 --> 00:41:41,589


1304
00:41:41,589 --> 00:41:43,990

 

1305
00:41:43,990 --> 00:41:46,069

 

1306
00:41:46,069 --> 00:41:47,510

 

1307
00:41:47,510 --> 00:41:49,510

 

1308
00:41:49,510 --> 00:41:52,230

 

1309
00:41:52,230 --> 00:41:54,470

 

1310
00:41:54,470 --> 00:41:58,230

 

1311
00:41:58,230 --> 00:42:01,190

 

1312
00:42:01,190 --> 00:42:03,349

 

1313
00:42:03,349 --> 00:42:05,510

 

1314
00:42:05,510 --> 00:42:08,230

 

1315
00:42:08,230 --> 00:42:10,390

 

1316
00:42:10,390 --> 00:42:12,150

 

1317
00:42:12,150 --> 00:42:14,790

 

1318
00:42:14,790 --> 00:42:17,270

 

1319
00:42:17,270 --> 00:42:18,710

 

1320
00:42:18,710 --> 00:42:18,720

 

1321
00:42:18,720 --> 00:42:20,069


1322
00:42:20,069 --> 00:42:20,079

 

1323
00:42:20,079 --> 00:42:21,030


1324
00:42:21,030 --> 00:42:23,270

 

1325
00:42:23,270 --> 00:42:24,630

 

1326
00:42:24,630 --> 00:42:24,640

 

1327
00:42:24,640 --> 00:42:29,670


1328
00:42:29,670 --> 00:42:34,069

 

1329
00:42:34,069 --> 00:42:34,079

 

1330
00:42:34,079 --> 00:42:34,870


1331
00:42:34,870 --> 00:42:37,349

 

1332
00:42:37,349 --> 00:42:39,270

 

1333
00:42:39,270 --> 00:42:42,150

 

1334
00:42:42,150 --> 00:42:43,589

 

1335
00:42:43,589 --> 00:42:44,630

 

1336
00:42:44,630 --> 00:42:44,640

 

1337
00:42:44,640 --> 00:42:46,069


1338
00:42:46,069 --> 00:42:47,430

 

1339
00:42:47,430 --> 00:42:49,030

 

1340
00:42:49,030 --> 00:42:51,589

 

1341
00:42:51,589 --> 00:42:54,630

 

1342
00:42:54,630 --> 00:42:56,550

 

1343
00:42:56,550 --> 00:42:58,550

 

1344
00:42:58,550 --> 00:43:00,069

 

1345
00:43:00,069 --> 00:43:00,079

 

1346
00:43:00,079 --> 00:43:01,829


1347
00:43:01,829 --> 00:43:01,839

 

1348
00:43:01,839 --> 00:43:05,109


1349
00:43:05,109 --> 00:43:05,119

 

1350
00:43:05,119 --> 00:43:06,230


1351
00:43:06,230 --> 00:43:06,240

 

1352
00:43:06,240 --> 00:43:07,829


1353
00:43:07,829 --> 00:43:11,030

 

1354
00:43:11,030 --> 00:43:13,030

 

1355
00:43:13,030 --> 00:43:14,550

 

1356
00:43:14,550 --> 00:43:16,069

 

1357
00:43:16,069 --> 00:43:17,910

 

1358
00:43:17,910 --> 00:43:19,670

 

1359
00:43:19,670 --> 00:43:19,680

 

1360
00:43:19,680 --> 00:43:20,950


1361
00:43:20,950 --> 00:43:23,270

 

1362
00:43:23,270 --> 00:43:25,430

 

1363
00:43:25,430 --> 00:43:28,950

 

1364
00:43:28,950 --> 00:43:32,390

 

1365
00:43:32,390 --> 00:43:35,829

 

1366
00:43:35,829 --> 00:43:37,349

 

1367
00:43:37,349 --> 00:43:37,359

 

1368
00:43:37,359 --> 00:43:39,430


1369
00:43:39,430 --> 00:43:40,870

 

1370
00:43:40,870 --> 00:43:43,750

 

1371
00:43:43,750 --> 00:43:45,349

 

1372
00:43:45,349 --> 00:43:47,109

 

1373
00:43:47,109 --> 00:43:50,069

 

1374
00:43:50,069 --> 00:43:51,750

 

1375
00:43:51,750 --> 00:43:53,430

 

1376
00:43:53,430 --> 00:43:54,950

 

1377
00:43:54,950 --> 00:43:56,870

 

1378
00:43:56,870 --> 00:44:00,309

 

1379
00:44:00,309 --> 00:44:01,910

 

1380
00:44:01,910 --> 00:44:01,920

 

1381
00:44:01,920 --> 00:44:03,910


1382
00:44:03,910 --> 00:44:05,349

 

1383
00:44:05,349 --> 00:44:07,910

 

1384
00:44:07,910 --> 00:44:10,230

 

1385
00:44:10,230 --> 00:44:12,870

 

1386
00:44:12,870 --> 00:44:15,510

 

1387
00:44:15,510 --> 00:44:18,309

 

1388
00:44:18,309 --> 00:44:22,550

 

1389
00:44:22,550 --> 00:44:22,560

 

1390
00:44:22,560 --> 00:44:23,430


1391
00:44:23,430 --> 00:44:25,990

 

1392
00:44:25,990 --> 00:44:27,670

 

1393
00:44:27,670 --> 00:44:29,190

 

1394
00:44:29,190 --> 00:44:29,200

 

1395
00:44:29,200 --> 00:44:30,069


1396
00:44:30,069 --> 00:44:32,230

 

1397
00:44:32,230 --> 00:44:33,829

 

1398
00:44:33,829 --> 00:44:37,430

 

1399
00:44:37,430 --> 00:44:39,510

 

1400
00:44:39,510 --> 00:44:39,520

 

1401
00:44:39,520 --> 00:44:40,630


1402
00:44:40,630 --> 00:44:40,640

 

1403
00:44:40,640 --> 00:44:41,430


1404
00:44:41,430 --> 00:44:43,990

 

1405
00:44:43,990 --> 00:44:46,470

 

1406
00:44:46,470 --> 00:44:46,480

 

1407
00:44:46,480 --> 00:44:47,990


1408
00:44:47,990 --> 00:44:50,550

 

1409
00:44:50,550 --> 00:44:52,470

 

1410
00:44:52,470 --> 00:44:54,230

 

1411
00:44:54,230 --> 00:44:56,710

 

1412
00:44:56,710 --> 00:44:59,510

 

1413
00:44:59,510 --> 00:44:59,520

 

1414
00:44:59,520 --> 00:45:00,790


1415
00:45:00,790 --> 00:45:02,630

 

1416
00:45:02,630 --> 00:45:04,790

 

1417
00:45:04,790 --> 00:45:04,800

 

1418
00:45:04,800 --> 00:45:07,270


1419
00:45:07,270 --> 00:45:09,030

 

1420
00:45:09,030 --> 00:45:10,710

 

1421
00:45:10,710 --> 00:45:13,430

 

1422
00:45:13,430 --> 00:45:15,190

 

1423
00:45:15,190 --> 00:45:17,750

 

1424
00:45:17,750 --> 00:45:18,950

 

1425
00:45:18,950 --> 00:45:22,150

 

1426
00:45:22,150 --> 00:45:24,470

 

1427
00:45:24,470 --> 00:45:26,950

 

1428
00:45:26,950 --> 00:45:29,990

 

1429
00:45:29,990 --> 00:45:32,790

 

1430
00:45:32,790 --> 00:45:34,309

 

1431
00:45:34,309 --> 00:45:34,319

 

1432
00:45:34,319 --> 00:45:36,309


1433
00:45:36,309 --> 00:45:39,510

 

1434
00:45:39,510 --> 00:45:41,270

 

1435
00:45:41,270 --> 00:45:42,950

 

1436
00:45:42,950 --> 00:45:44,390

 

1437
00:45:44,390 --> 00:45:48,150

 

1438
00:45:48,150 --> 00:45:52,150

 

1439
00:45:52,150 --> 00:45:54,550

 

1440
00:45:54,550 --> 00:45:56,150

 

1441
00:45:56,150 --> 00:45:59,430

 

1442
00:45:59,430 --> 00:45:59,440

 

1443
00:45:59,440 --> 00:46:02,710


1444
00:46:02,710 --> 00:46:02,720

 

1445
00:46:02,720 --> 00:46:05,750


1446
00:46:05,750 --> 00:46:05,760

 

1447
00:46:05,760 --> 00:46:06,550


1448
00:46:06,550 --> 00:46:06,560

 

1449
00:46:06,560 --> 00:46:08,150


1450
00:46:08,150 --> 00:46:10,150

 

1451
00:46:10,150 --> 00:46:10,160

 

1452
00:46:10,160 --> 00:46:11,190


1453
00:46:11,190 --> 00:46:13,349

 

1454
00:46:13,349 --> 00:46:15,829

 

1455
00:46:15,829 --> 00:46:17,750

 

1456
00:46:17,750 --> 00:46:19,670

 

1457
00:46:19,670 --> 00:46:20,950

 

1458
00:46:20,950 --> 00:46:20,960

 

1459
00:46:20,960 --> 00:46:21,750


1460
00:46:21,750 --> 00:46:23,510

 

1461
00:46:23,510 --> 00:46:26,710

 

1462
00:46:26,710 --> 00:46:26,720

 

1463
00:46:26,720 --> 00:46:27,750


1464
00:46:27,750 --> 00:46:30,150

 

1465
00:46:30,150 --> 00:46:31,510

 

1466
00:46:31,510 --> 00:46:33,589

 

1467
00:46:33,589 --> 00:46:35,270

 

1468
00:46:35,270 --> 00:46:37,589

 

1469
00:46:37,589 --> 00:46:39,670

 

1470
00:46:39,670 --> 00:46:42,390

 

1471
00:46:42,390 --> 00:46:45,030

 

1472
00:46:45,030 --> 00:46:46,710

 

1473
00:46:46,710 --> 00:46:49,030

 

1474
00:46:49,030 --> 00:46:51,109

 

1475
00:46:51,109 --> 00:46:52,790

 

1476
00:46:52,790 --> 00:46:54,790

 

1477
00:46:54,790 --> 00:46:57,430

 

1478
00:46:57,430 --> 00:46:58,950

 

1479
00:46:58,950 --> 00:47:01,109

 

1480
00:47:01,109 --> 00:47:03,349

 

1481
00:47:03,349 --> 00:47:04,390

 

1482
00:47:04,390 --> 00:47:06,870

 

1483
00:47:06,870 --> 00:47:10,390

 

1484
00:47:10,390 --> 00:47:11,589

 

1485
00:47:11,589 --> 00:47:11,599

 

1486
00:47:11,599 --> 00:47:12,790


1487
00:47:12,790 --> 00:47:14,630

 

1488
00:47:14,630 --> 00:47:16,950

 

1489
00:47:16,950 --> 00:47:19,430

 

1490
00:47:19,430 --> 00:47:21,910

 

1491
00:47:21,910 --> 00:47:24,230

 

1492
00:47:24,230 --> 00:47:25,670

 

1493
00:47:25,670 --> 00:47:27,430

 

1494
00:47:27,430 --> 00:47:29,030

 

1495
00:47:29,030 --> 00:47:29,040

 

1496
00:47:29,040 --> 00:47:30,309


1497
00:47:30,309 --> 00:47:32,710

 

1498
00:47:32,710 --> 00:47:32,720

 

1499
00:47:32,720 --> 00:47:33,829


1500
00:47:33,829 --> 00:47:35,270

 

1501
00:47:35,270 --> 00:47:37,510

 

1502
00:47:37,510 --> 00:47:40,390

 

1503
00:47:40,390 --> 00:47:40,400

 

1504
00:47:40,400 --> 00:47:42,950


1505
00:47:42,950 --> 00:47:44,630

 

1506
00:47:44,630 --> 00:47:46,710

 

1507
00:47:46,710 --> 00:47:48,390

 

1508
00:47:48,390 --> 00:47:50,069

 

1509
00:47:50,069 --> 00:47:55,349

 

1510
00:47:55,349 --> 00:47:57,030

 

1511
00:47:57,030 --> 00:47:59,750

 

1512
00:47:59,750 --> 00:48:03,109

 

1513
00:48:03,109 --> 00:48:06,309

 

1514
00:48:06,309 --> 00:48:08,630

 

1515
00:48:08,630 --> 00:48:12,069

 

1516
00:48:12,069 --> 00:48:14,309

 

1517
00:48:14,309 --> 00:48:14,319

 

1518
00:48:14,319 --> 00:48:18,309


1519
00:48:18,309 --> 00:48:21,030

 

1520
00:48:21,030 --> 00:48:22,470

 

1521
00:48:22,470 --> 00:48:25,750

 

1522
00:48:25,750 --> 00:48:27,510

 

1523
00:48:27,510 --> 00:48:29,829

 

1524
00:48:29,829 --> 00:48:29,839

 

1525
00:48:29,839 --> 00:48:31,910


1526
00:48:31,910 --> 00:48:35,270

 

1527
00:48:35,270 --> 00:48:37,190

 

1528
00:48:37,190 --> 00:48:39,430

 

1529
00:48:39,430 --> 00:48:39,440

 

1530
00:48:39,440 --> 00:48:44,069


1531
00:48:44,069 --> 00:48:45,829

 

1532
00:48:45,829 --> 00:48:48,630

 

1533
00:48:48,630 --> 00:48:50,630

 

1534
00:48:50,630 --> 00:48:50,640

 

1535
00:48:50,640 --> 00:48:52,150


1536
00:48:52,150 --> 00:48:56,069

 

1537
00:48:56,069 --> 00:49:01,109

 

1538
00:49:01,109 --> 00:49:03,430

 

1539
00:49:03,430 --> 00:49:03,440

 

1540
00:49:03,440 --> 00:49:04,470


1541
00:49:04,470 --> 00:49:06,230

 

1542
00:49:06,230 --> 00:49:09,990

 

1543
00:49:09,990 --> 00:49:12,150

 

1544
00:49:12,150 --> 00:49:14,390

 

1545
00:49:14,390 --> 00:49:14,400

 

1546
00:49:14,400 --> 00:49:16,470


1547
00:49:16,470 --> 00:49:18,549

 

1548
00:49:18,549 --> 00:49:19,750

 

1549
00:49:19,750 --> 00:49:21,670

 

1550
00:49:21,670 --> 00:49:24,309

 

1551
00:49:24,309 --> 00:49:25,910

 

1552
00:49:25,910 --> 00:49:27,030

 

1553
00:49:27,030 --> 00:49:29,030

 

1554
00:49:29,030 --> 00:49:30,549

 

1555
00:49:30,549 --> 00:49:31,829

 

1556
00:49:31,829 --> 00:49:35,430

 

1557
00:49:35,430 --> 00:49:37,349

 

1558
00:49:37,349 --> 00:49:37,359

 

1559
00:49:37,359 --> 00:49:38,950


1560
00:49:38,950 --> 00:49:40,470

 

1561
00:49:40,470 --> 00:49:42,870

 

1562
00:49:42,870 --> 00:49:45,750

 

1563
00:49:45,750 --> 00:49:47,430

 

1564
00:49:47,430 --> 00:49:50,950

 

1565
00:49:50,950 --> 00:49:55,190

 

1566
00:49:55,190 --> 00:49:55,200

 

1567
00:49:55,200 --> 00:49:57,109


1568
00:49:57,109 --> 00:50:00,230

 

1569
00:50:00,230 --> 00:50:02,470

 

1570
00:50:02,470 --> 00:50:05,349

 

1571
00:50:05,349 --> 00:50:06,870

 

1572
00:50:06,870 --> 00:50:08,390

 

1573
00:50:08,390 --> 00:50:10,390

 

1574
00:50:10,390 --> 00:50:11,910

 

1575
00:50:11,910 --> 00:50:11,920

 

1576
00:50:11,920 --> 00:50:12,950


1577
00:50:12,950 --> 00:50:14,549

 

1578
00:50:14,549 --> 00:50:17,109

 

1579
00:50:17,109 --> 00:50:18,950

 

1580
00:50:18,950 --> 00:50:18,960

 

1581
00:50:18,960 --> 00:50:20,390


1582
00:50:20,390 --> 00:50:21,910

 

1583
00:50:21,910 --> 00:50:23,910

 

1584
00:50:23,910 --> 00:50:26,549

 

1585
00:50:26,549 --> 00:50:28,950

 

1586
00:50:28,950 --> 00:50:28,960

 

1587
00:50:28,960 --> 00:50:30,470


1588
00:50:30,470 --> 00:50:32,230

 

1589
00:50:32,230 --> 00:50:33,670

 

1590
00:50:33,670 --> 00:50:35,990

 

1591
00:50:35,990 --> 00:50:38,230

 

1592
00:50:38,230 --> 00:50:40,470

 

1593
00:50:40,470 --> 00:50:42,309

 

1594
00:50:42,309 --> 00:50:42,319

 

1595
00:50:42,319 --> 00:50:43,510


1596
00:50:43,510 --> 00:50:45,750

 

1597
00:50:45,750 --> 00:50:48,309

 

1598
00:50:48,309 --> 00:50:50,470

 

1599
00:50:50,470 --> 00:50:52,309

 

1600
00:50:52,309 --> 00:50:53,670

 

1601
00:50:53,670 --> 00:50:56,230

 

1602
00:50:56,230 --> 00:50:58,150

 

1603
00:50:58,150 --> 00:50:58,160

 

1604
00:50:58,160 --> 00:50:58,950


1605
00:50:58,950 --> 00:50:59,829

 

1606
00:50:59,829 --> 00:51:01,270

 

1607
00:51:01,270 --> 00:51:03,670

 

1608
00:51:03,670 --> 00:51:04,870

 

1609
00:51:04,870 --> 00:51:06,870

 

1610
00:51:06,870 --> 00:51:11,270

 

1611
00:51:11,270 --> 00:51:13,349

 

1612
00:51:13,349 --> 00:51:15,190

 

1613
00:51:15,190 --> 00:51:17,990

 

1614
00:51:17,990 --> 00:51:20,549

 

1615
00:51:20,549 --> 00:51:22,950

 

1616
00:51:22,950 --> 00:51:24,230

 

1617
00:51:24,230 --> 00:51:26,710

 

1618
00:51:26,710 --> 00:51:26,720

 

1619
00:51:26,720 --> 00:51:29,589


1620
00:51:29,589 --> 00:51:31,589

 

1621
00:51:31,589 --> 00:51:33,670

 

1622
00:51:33,670 --> 00:51:34,790

 

1623
00:51:34,790 --> 00:51:37,109

 

1624
00:51:37,109 --> 00:51:39,670

 

1625
00:51:39,670 --> 00:51:50,549

 

1626
00:51:50,549 --> 00:51:53,030

 

1627
00:51:53,030 --> 00:51:53,040

 

1628
00:51:53,040 --> 00:51:53,829


1629
00:51:53,829 --> 00:51:55,349

 

1630
00:51:55,349 --> 00:51:57,430

 

1631
00:51:57,430 --> 00:52:00,150

 

1632
00:52:00,150 --> 00:52:01,829

 

1633
00:52:01,829 --> 00:52:01,839

 

1634
00:52:01,839 --> 00:52:03,829


1635
00:52:03,829 --> 00:52:06,950

 

1636
00:52:06,950 --> 00:52:07,990

 

1637
00:52:07,990 --> 00:52:09,670

 

1638
00:52:09,670 --> 00:52:10,870

 

1639
00:52:10,870 --> 00:52:12,710

 

1640
00:52:12,710 --> 00:52:15,270

 

1641
00:52:15,270 --> 00:52:17,190

 

1642
00:52:17,190 --> 00:52:18,470

 

1643
00:52:18,470 --> 00:52:21,349

 

1644
00:52:21,349 --> 00:52:22,630

 

1645
00:52:22,630 --> 00:52:25,430

 

1646
00:52:25,430 --> 00:52:27,829

 

1647
00:52:27,829 --> 00:52:29,270

 

1648
00:52:29,270 --> 00:52:30,549

 

1649
00:52:30,549 --> 00:52:32,309

 

1650
00:52:32,309 --> 00:52:33,910

 

1651
00:52:33,910 --> 00:52:36,309

 

1652
00:52:36,309 --> 00:52:36,319

 

1653
00:52:36,319 --> 00:52:37,670


1654
00:52:37,670 --> 00:52:38,710

 

1655
00:52:38,710 --> 00:52:40,950

 

1656
00:52:40,950 --> 00:52:42,630

 

1657
00:52:42,630 --> 00:52:44,630

 

1658
00:52:44,630 --> 00:52:46,230

 

1659
00:52:46,230 --> 00:52:48,549

 

1660
00:52:48,549 --> 00:52:51,270

 

1661
00:52:51,270 --> 00:52:51,280

 

1662
00:52:51,280 --> 00:52:53,030


1663
00:52:53,030 --> 00:52:55,030

 

1664
00:52:55,030 --> 00:52:56,870

 

1665
00:52:56,870 --> 00:52:58,829

 

1666
00:52:58,829 --> 00:53:00,870

 

1667
00:53:00,870 --> 00:53:03,430

 

1668
00:53:03,430 --> 00:53:07,510

 

1669
00:53:07,510 --> 00:53:09,190

 

1670
00:53:09,190 --> 00:53:11,270

 

1671
00:53:11,270 --> 00:53:12,870

 

1672
00:53:12,870 --> 00:53:14,309

 

1673
00:53:14,309 --> 00:53:15,750

 

1674
00:53:15,750 --> 00:53:19,109

 

1675
00:53:19,109 --> 00:53:20,390

 

1676
00:53:20,390 --> 00:53:22,549

 

1677
00:53:22,549 --> 00:53:24,790

 

1678
00:53:24,790 --> 00:53:24,800

 

1679
00:53:24,800 --> 00:53:25,420


1680
00:53:25,420 --> 00:53:25,430

 

1681
00:53:25,430 --> 00:53:26,870


1682
00:53:26,870 --> 00:53:26,880

 

1683
00:53:26,880 --> 00:53:27,670


1684
00:53:27,670 --> 00:53:28,710

 

1685
00:53:28,710 --> 00:53:30,390

 

1686
00:53:30,390 --> 00:53:32,309

 

1687
00:53:32,309 --> 00:53:34,309

 

1688
00:53:34,309 --> 00:53:36,390

 

1689
00:53:36,390 --> 00:53:38,950

 

1690
00:53:38,950 --> 00:53:40,950

 

1691
00:53:40,950 --> 00:53:40,960

 

1692
00:53:40,960 --> 00:53:42,390


1693
00:53:42,390 --> 00:53:43,750

 

1694
00:53:43,750 --> 00:53:47,190

 

1695
00:53:47,190 --> 00:53:52,069

 

1696
00:53:52,069 --> 00:53:54,230

 

1697
00:53:54,230 --> 00:53:55,510

 

1698
00:53:55,510 --> 00:53:55,520

 

1699
00:53:55,520 --> 00:53:56,710


1700
00:53:56,710 --> 00:53:59,430

 

1701
00:53:59,430 --> 00:54:01,349

 

1702
00:54:01,349 --> 00:54:03,910

 

1703
00:54:03,910 --> 00:54:06,390

 

1704
00:54:06,390 --> 00:54:09,030

 

1705
00:54:09,030 --> 00:54:09,040

 

1706
00:54:09,040 --> 00:54:13,109


1707
00:54:13,109 --> 00:54:15,349

 

1708
00:54:15,349 --> 00:54:17,670

 

1709
00:54:17,670 --> 00:54:20,470

 

1710
00:54:20,470 --> 00:54:21,589

 

1711
00:54:21,589 --> 00:54:23,589

 

1712
00:54:23,589 --> 00:54:26,309

 

1713
00:54:26,309 --> 00:54:27,829

 

1714
00:54:27,829 --> 00:54:30,870

 

1715
00:54:30,870 --> 00:54:33,829

 

1716
00:54:33,829 --> 00:54:35,990

 

1717
00:54:35,990 --> 00:54:40,309

 

1718
00:54:40,309 --> 00:54:41,829

 

1719
00:54:41,829 --> 00:54:43,109

 

1720
00:54:43,109 --> 00:54:43,119

 

1721
00:54:43,119 --> 00:54:44,069


1722
00:54:44,069 --> 00:54:46,549

 

1723
00:54:46,549 --> 00:54:46,559

 

1724
00:54:46,559 --> 00:54:48,150


1725
00:54:48,150 --> 00:54:49,910

 

1726
00:54:49,910 --> 00:54:49,920

 

1727
00:54:49,920 --> 00:54:52,549


1728
00:54:52,549 --> 00:54:55,750

 

1729
00:54:55,750 --> 00:54:55,760

 

1730
00:54:55,760 --> 00:54:56,630


1731
00:54:56,630 --> 00:54:56,640

 

1732
00:54:56,640 --> 00:54:58,069


1733
00:54:58,069 --> 00:55:00,230

 

1734
00:55:00,230 --> 00:55:01,990

 

1735
00:55:01,990 --> 00:55:03,190

 

1736
00:55:03,190 --> 00:55:05,510

 

1737
00:55:05,510 --> 00:55:07,829

 

1738
00:55:07,829 --> 00:55:10,870

 

1739
00:55:10,870 --> 00:55:12,870

 

1740
00:55:12,870 --> 00:55:14,549

 

1741
00:55:14,549 --> 00:55:17,030

 

1742
00:55:17,030 --> 00:55:18,630

 

1743
00:55:18,630 --> 00:55:20,230

 

1744
00:55:20,230 --> 00:55:23,109

 

1745
00:55:23,109 --> 00:55:26,710

 

1746
00:55:26,710 --> 00:55:26,720

 

1747
00:55:26,720 --> 00:55:27,910


1748
00:55:27,910 --> 00:55:30,470

 

1749
00:55:30,470 --> 00:55:32,630

 

1750
00:55:32,630 --> 00:55:36,390

 

1751
00:55:36,390 --> 00:55:37,670

 

1752
00:55:37,670 --> 00:55:39,990

 

1753
00:55:39,990 --> 00:55:43,670

 

1754
00:55:43,670 --> 00:55:45,030

 

1755
00:55:45,030 --> 00:55:47,109

 

1756
00:55:47,109 --> 00:55:47,119

 

1757
00:55:47,119 --> 00:55:48,710


1758
00:55:48,710 --> 00:55:51,910

 

1759
00:55:51,910 --> 00:55:51,920

 

1760
00:55:51,920 --> 00:55:53,750


1761
00:55:53,750 --> 00:55:53,760

 

1762
00:55:53,760 --> 00:55:55,349


1763
00:55:55,349 --> 00:55:58,789

 

1764
00:55:58,789 --> 00:56:01,030

 

1765
00:56:01,030 --> 00:56:02,549

 

1766
00:56:02,549 --> 00:56:04,069

 

1767
00:56:04,069 --> 00:56:05,829

 

1768
00:56:05,829 --> 00:56:08,390

 

1769
00:56:08,390 --> 00:56:12,069

 

1770
00:56:12,069 --> 00:56:14,630

 

1771
00:56:14,630 --> 00:56:18,390

 

1772
00:56:18,390 --> 00:56:20,710

 

1773
00:56:20,710 --> 00:56:22,069

 

1774
00:56:22,069 --> 00:56:22,079

 

1775
00:56:22,079 --> 00:56:23,030


1776
00:56:23,030 --> 00:56:25,750

 

1777
00:56:25,750 --> 00:56:31,030

 

1778
00:56:31,030 --> 00:56:36,470

 

1779
00:56:36,470 --> 00:56:38,630

 

1780
00:56:38,630 --> 00:56:42,870

 

1781
00:56:42,870 --> 00:56:46,069

 

1782
00:56:46,069 --> 00:56:49,190

 

1783
00:56:49,190 --> 00:56:50,789

 

1784
00:56:50,789 --> 00:56:52,150

 

1785
00:56:52,150 --> 00:56:53,829

 

1786
00:56:53,829 --> 00:56:53,839

 

1787
00:56:53,839 --> 00:56:54,870


1788
00:56:54,870 --> 00:56:56,950

 

1789
00:56:56,950 --> 00:57:00,390

 

1790
00:57:00,390 --> 00:57:00,400

 

1791
00:57:00,400 --> 00:57:03,030


1792
00:57:03,030 --> 00:57:05,030

 

1793
00:57:05,030 --> 00:57:06,870

 

1794
00:57:06,870 --> 00:57:10,230

 

1795
00:57:10,230 --> 00:57:10,240

 

1796
00:57:10,240 --> 00:57:13,190


1797
00:57:13,190 --> 00:57:15,990

 

1798
00:57:15,990 --> 00:57:18,549

 

1799
00:57:18,549 --> 00:57:18,559

 

1800
00:57:18,559 --> 00:57:19,510


1801
00:57:19,510 --> 00:57:23,670

 

1802
00:57:23,670 --> 00:57:26,630

 

1803
00:57:26,630 --> 00:57:27,670

 

1804
00:57:27,670 --> 00:57:30,950

 

1805
00:57:30,950 --> 00:57:35,510

 

1806
00:57:35,510 --> 00:57:37,430

 

1807
00:57:37,430 --> 00:57:37,440

 

1808
00:57:37,440 --> 00:57:38,630


1809
00:57:38,630 --> 00:57:41,030

 

1810
00:57:41,030 --> 00:57:43,109

 

1811
00:57:43,109 --> 00:57:44,789

 

1812
00:57:44,789 --> 00:57:44,799

 

1813
00:57:44,799 --> 00:57:45,750


1814
00:57:45,750 --> 00:57:47,910

 

1815
00:57:47,910 --> 00:57:50,069

 

1816
00:57:50,069 --> 00:57:52,069

 

1817
00:57:52,069 --> 00:57:53,990

 

1818
00:57:53,990 --> 00:57:56,710

 

1819
00:57:56,710 --> 00:57:59,030

 

1820
00:57:59,030 --> 00:58:01,510

 

1821
00:58:01,510 --> 00:58:03,589

 

1822
00:58:03,589 --> 00:58:06,789

 

1823
00:58:06,789 --> 00:58:08,950

 

1824
00:58:08,950 --> 00:58:11,270

 

1825
00:58:11,270 --> 00:58:13,829

 

1826
00:58:13,829 --> 00:58:17,829

 

1827
00:58:17,829 --> 00:58:17,839

 

1828
00:58:17,839 --> 00:58:19,030


1829
00:58:19,030 --> 00:58:21,190

 

1830
00:58:21,190 --> 00:58:21,200

 

1831
00:58:21,200 --> 00:58:22,549


1832
00:58:22,549 --> 00:58:22,559

 

1833
00:58:22,559 --> 00:58:23,430


1834
00:58:23,430 --> 00:58:25,270

 

1835
00:58:25,270 --> 00:58:27,030

 

1836
00:58:27,030 --> 00:58:29,349

 

1837
00:58:29,349 --> 00:58:29,359

 

1838
00:58:29,359 --> 00:58:32,230


1839
00:58:32,230 --> 00:58:35,270

 

1840
00:58:35,270 --> 00:58:37,910

 

1841
00:58:37,910 --> 00:58:43,990

 

1842
00:58:43,990 --> 00:58:45,589

 

1843
00:58:45,589 --> 00:58:47,829

 

1844
00:58:47,829 --> 00:58:49,349

 

1845
00:58:49,349 --> 00:58:52,710

 

1846
00:58:52,710 --> 00:58:55,270

 

1847
00:58:55,270 --> 00:58:55,280

 

1848
00:58:55,280 --> 00:58:57,190


1849
00:58:57,190 --> 00:58:59,030

 

1850
00:58:59,030 --> 00:59:00,390

 

1851
00:59:00,390 --> 00:59:03,670

 

1852
00:59:03,670 --> 00:59:05,510

 

1853
00:59:05,510 --> 00:59:06,870

 

1854
00:59:06,870 --> 00:59:06,880

 

1855
00:59:06,880 --> 00:59:09,670


1856
00:59:09,670 --> 00:59:12,069

 

1857
00:59:12,069 --> 00:59:15,910

 

1858
00:59:15,910 --> 00:59:21,670

 

1859
00:59:21,670 --> 00:59:23,589

 

1860
00:59:23,589 --> 00:59:25,430

 

1861
00:59:25,430 --> 00:59:27,190

 

1862
00:59:27,190 --> 00:59:27,200

 

1863
00:59:27,200 --> 00:59:28,069


1864
00:59:28,069 --> 00:59:29,349

 

1865
00:59:29,349 --> 00:59:29,359

 

1866
00:59:29,359 --> 00:59:30,870


1867
00:59:30,870 --> 00:59:33,589

 

1868
00:59:33,589 --> 00:59:35,190

 

1869
00:59:35,190 --> 00:59:37,109

 

1870
00:59:37,109 --> 00:59:39,510

 

1871
00:59:39,510 --> 00:59:42,150

 

1872
00:59:42,150 --> 00:59:43,270

 

1873
00:59:43,270 --> 00:59:45,510

 

1874
00:59:45,510 --> 00:59:46,950

 

1875
00:59:46,950 --> 00:59:48,390

 

1876
00:59:48,390 --> 00:59:49,750

 

1877
00:59:49,750 --> 00:59:51,750

 

1878
00:59:51,750 --> 00:59:53,510

 

1879
00:59:53,510 --> 00:59:55,349

 

1880
00:59:55,349 --> 00:59:56,390

 

1881
00:59:56,390 --> 00:59:59,190

 

1882
00:59:59,190 --> 01:00:01,430

 

1883
01:00:01,430 --> 01:00:04,549

 

1884
01:00:04,549 --> 01:00:04,559

 

1885
01:00:04,559 --> 01:00:05,510


1886
01:00:05,510 --> 01:00:06,789

 

1887
01:00:06,789 --> 01:00:08,710

 

1888
01:00:08,710 --> 01:00:10,309

 

1889
01:00:10,309 --> 01:00:12,309

 

1890
01:00:12,309 --> 01:00:14,789

 

1891
01:00:14,789 --> 01:00:16,069

 

1892
01:00:16,069 --> 01:00:18,150

 

1893
01:00:18,150 --> 01:00:19,349

 

1894
01:00:19,349 --> 01:00:19,359

 

1895
01:00:19,359 --> 01:00:20,789


1896
01:00:20,789 --> 01:00:23,109

 

1897
01:00:23,109 --> 01:00:25,589

 

1898
01:00:25,589 --> 01:00:27,750

 

1899
01:00:27,750 --> 01:00:29,750

 

1900
01:00:29,750 --> 01:00:31,109

 

1901
01:00:31,109 --> 01:00:34,950

 

1902
01:00:34,950 --> 01:00:37,109

 

1903
01:00:37,109 --> 01:00:39,190

 

1904
01:00:39,190 --> 01:00:41,109

 

1905
01:00:41,109 --> 01:00:43,270

 

1906
01:00:43,270 --> 01:00:45,910

 

1907
01:00:45,910 --> 01:00:48,710

 

1908
01:00:48,710 --> 01:00:50,630

 

1909
01:00:50,630 --> 01:00:52,630

 

1910
01:00:52,630 --> 01:00:55,109

 

1911
01:00:55,109 --> 01:00:56,630

 

1912
01:00:56,630 --> 01:00:58,630

 

1913
01:00:58,630 --> 01:01:00,470

 

1914
01:01:00,470 --> 01:01:00,480

 

1915
01:01:00,480 --> 01:01:01,750


1916
01:01:01,750 --> 01:01:01,760

 

1917
01:01:01,760 --> 01:01:02,630


1918
01:01:02,630 --> 01:01:06,230

 

1919
01:01:06,230 --> 01:01:08,069

 

1920
01:01:08,069 --> 01:01:09,430

 

1921
01:01:09,430 --> 01:01:11,750

 

1922
01:01:11,750 --> 01:01:13,990

 

1923
01:01:13,990 --> 01:01:14,000

 

1924
01:01:14,000 --> 01:01:14,870


1925
01:01:14,870 --> 01:01:16,390

 

1926
01:01:16,390 --> 01:01:18,470

 

1927
01:01:18,470 --> 01:01:20,950

 

1928
01:01:20,950 --> 01:01:23,430

 

1929
01:01:23,430 --> 01:01:24,630

 

1930
01:01:24,630 --> 01:01:27,910

 

1931
01:01:27,910 --> 01:01:27,920

 

1932
01:01:27,920 --> 01:01:30,549


1933
01:01:30,549 --> 01:01:36,230

 

1934
01:01:36,230 --> 01:01:37,430

 

1935
01:01:37,430 --> 01:01:39,750

 

1936
01:01:39,750 --> 01:01:42,069

 

1937
01:01:42,069 --> 01:01:43,510

 

1938
01:01:43,510 --> 01:01:43,520

 

1939
01:01:43,520 --> 01:01:44,789


1940
01:01:44,789 --> 01:01:46,950

 

1941
01:01:46,950 --> 01:01:48,390

 

1942
01:01:48,390 --> 01:01:50,710

 

1943
01:01:50,710 --> 01:01:52,390

 

1944
01:01:52,390 --> 01:01:55,589

 

1945
01:01:55,589 --> 01:01:55,599

 

1946
01:01:55,599 --> 01:01:58,150


1947
01:01:58,150 --> 01:01:58,160

 

1948
01:01:58,160 --> 01:01:59,750


1949
01:01:59,750 --> 01:02:02,870

 

1950
01:02:02,870 --> 01:02:05,670

 

1951
01:02:05,670 --> 01:02:06,950

 

1952
01:02:06,950 --> 01:02:09,349

 

1953
01:02:09,349 --> 01:02:11,990

 

1954
01:02:11,990 --> 01:02:12,000

 

1955
01:02:12,000 --> 01:02:14,870


1956
01:02:14,870 --> 01:02:14,880

 

1957
01:02:14,880 --> 01:02:15,990


1958
01:02:15,990 --> 01:02:17,670

 

1959
01:02:17,670 --> 01:02:19,990

 

1960
01:02:19,990 --> 01:02:22,150

 

1961
01:02:22,150 --> 01:02:23,589

 

1962
01:02:23,589 --> 01:02:25,029

 

1963
01:02:25,029 --> 01:02:26,870

 

1964
01:02:26,870 --> 01:02:28,549

 

1965
01:02:28,549 --> 01:02:31,829

 

1966
01:02:31,829 --> 01:02:33,589

 

1967
01:02:33,589 --> 01:02:33,599

 

1968
01:02:33,599 --> 01:02:35,029


1969
01:02:35,029 --> 01:02:37,349

 

1970
01:02:37,349 --> 01:02:40,789

 

1971
01:02:40,789 --> 01:02:44,150

 

1972
01:02:44,150 --> 01:02:46,710

 

1973
01:02:46,710 --> 01:02:49,829

 

1974
01:02:49,829 --> 01:02:49,839

 

1975
01:02:49,839 --> 01:02:51,750


1976
01:02:51,750 --> 01:02:55,190

 

1977
01:02:55,190 --> 01:02:55,200

 

1978
01:02:55,200 --> 01:02:56,230


1979
01:02:56,230 --> 01:02:58,630

 

1980
01:02:58,630 --> 01:02:58,640

 

1981
01:02:58,640 --> 01:03:00,630


1982
01:03:00,630 --> 01:03:01,750

 

1983
01:03:01,750 --> 01:03:03,589

 

1984
01:03:03,589 --> 01:03:06,390

 

1985
01:03:06,390 --> 01:03:10,390

 

1986
01:03:10,390 --> 01:03:12,789

 

1987
01:03:12,789 --> 01:03:12,799

 

1988
01:03:12,799 --> 01:03:13,750


1989
01:03:13,750 --> 01:03:14,829

 

1990
01:03:14,829 --> 01:03:14,839

 

1991
01:03:14,839 --> 01:03:16,549


1992
01:03:16,549 --> 01:03:19,910

 

1993
01:03:19,910 --> 01:03:19,920

 

1994
01:03:19,920 --> 01:03:21,190


1995
01:03:21,190 --> 01:03:24,069

 

1996
01:03:24,069 --> 01:03:24,079

 

1997
01:03:24,079 --> 01:03:25,430


1998
01:03:25,430 --> 01:03:27,430

 

1999
01:03:27,430 --> 01:03:34,870

 

2000
01:03:34,870 --> 01:03:37,029

 

2001
01:03:37,029 --> 01:03:40,549

 

2002
01:03:40,549 --> 01:03:43,349

 

2003
01:03:43,349 --> 01:03:45,270

 

2004
01:03:45,270 --> 01:03:49,029

 

2005
01:03:49,029 --> 01:03:49,039

 

2006
01:03:49,039 --> 01:03:50,230


2007
01:03:50,230 --> 01:03:53,270

 

2008
01:03:53,270 --> 01:03:55,670

 

2009
01:03:55,670 --> 01:03:57,430

 

2010
01:03:57,430 --> 01:03:59,750

 

2011
01:03:59,750 --> 01:04:01,029

 

2012
01:04:01,029 --> 01:04:03,750

 

2013
01:04:03,750 --> 01:04:05,430

 

2014
01:04:05,430 --> 01:04:07,190

 

2015
01:04:07,190 --> 01:04:08,950

 

2016
01:04:08,950 --> 01:04:11,270

 

2017
01:04:11,270 --> 01:04:13,029

 

2018
01:04:13,029 --> 01:04:15,190

 

2019
01:04:15,190 --> 01:04:18,789

 

2020
01:04:18,789 --> 01:04:20,470

 

2021
01:04:20,470 --> 01:04:23,750

 

2022
01:04:23,750 --> 01:04:26,710

 

2023
01:04:26,710 --> 01:04:31,990

 

2024
01:04:31,990 --> 01:04:33,750

 

2025
01:04:33,750 --> 01:04:36,230

 

2026
01:04:36,230 --> 01:04:36,240

 

2027
01:04:36,240 --> 01:04:37,029


2028
01:04:37,029 --> 01:04:39,109

 

2029
01:04:39,109 --> 01:04:39,119

 

2030
01:04:39,119 --> 01:04:42,230


2031
01:04:42,230 --> 01:04:46,069

 

2032
01:04:46,069 --> 01:04:48,950

 

2033
01:04:48,950 --> 01:04:51,829

 

2034
01:04:51,829 --> 01:04:56,069

 

2035
01:04:56,069 --> 01:04:57,589

 

2036
01:04:57,589 --> 01:04:59,029

 

2037
01:04:59,029 --> 01:04:59,039

 

2038
01:04:59,039 --> 01:05:01,190


2039
01:05:01,190 --> 01:05:03,510

 

2040
01:05:03,510 --> 01:05:05,190

 

2041
01:05:05,190 --> 01:05:07,430

 

2042
01:05:07,430 --> 01:05:09,510

 

2043
01:05:09,510 --> 01:05:12,390

 

2044
01:05:12,390 --> 01:05:13,910

 

2045
01:05:13,910 --> 01:05:15,750

 

2046
01:05:15,750 --> 01:05:17,990

 

2047
01:05:17,990 --> 01:05:19,349

 

2048
01:05:19,349 --> 01:05:19,359

 

2049
01:05:19,359 --> 01:05:20,870


2050
01:05:20,870 --> 01:05:22,309

 

2051
01:05:22,309 --> 01:05:26,630

 

2052
01:05:26,630 --> 01:05:28,630

 

2053
01:05:28,630 --> 01:05:30,789

 

2054
01:05:30,789 --> 01:05:33,109

 

2055
01:05:33,109 --> 01:05:35,270

 

2056
01:05:35,270 --> 01:05:39,829

 

2057
01:05:39,829 --> 01:05:41,910

 

2058
01:05:41,910 --> 01:05:44,710

 

2059
01:05:44,710 --> 01:05:46,470

 

2060
01:05:46,470 --> 01:05:49,430

 

2061
01:05:49,430 --> 01:05:51,029

 

2062
01:05:51,029 --> 01:05:51,039

 

2063
01:05:51,039 --> 01:05:52,470


2064
01:05:52,470 --> 01:05:53,510

 

2065
01:05:53,510 --> 01:05:54,950

 

2066
01:05:54,950 --> 01:05:56,710

 

2067
01:05:56,710 --> 01:05:59,910

 

2068
01:05:59,910 --> 01:06:01,430

 

2069
01:06:01,430 --> 01:06:13,430

 

2070
01:06:13,430 --> 01:06:15,349

 

2071
01:06:15,349 --> 01:06:17,190

 

2072
01:06:17,190 --> 01:06:18,390

 

2073
01:06:18,390 --> 01:06:20,069

 

2074
01:06:20,069 --> 01:06:22,069

 

2075
01:06:22,069 --> 01:06:24,150

 

2076
01:06:24,150 --> 01:06:24,160

 

2077
01:06:24,160 --> 01:06:25,510


2078
01:06:25,510 --> 01:06:28,470

 

2079
01:06:28,470 --> 01:06:29,829

 

2080
01:06:29,829 --> 01:06:32,309

 

2081
01:06:32,309 --> 01:06:34,150

 

2082
01:06:34,150 --> 01:06:35,670

 

2083
01:06:35,670 --> 01:06:37,750

 

2084
01:06:37,750 --> 01:06:37,760

 

2085
01:06:37,760 --> 01:06:39,510


2086
01:06:39,510 --> 01:06:41,190

 

2087
01:06:41,190 --> 01:06:42,950

 

2088
01:06:42,950 --> 01:06:44,870

 

2089
01:06:44,870 --> 01:06:46,069

 

2090
01:06:46,069 --> 01:06:47,270

 

2091
01:06:47,270 --> 01:06:48,710

 

2092
01:06:48,710 --> 01:06:50,630

 

2093
01:06:50,630 --> 01:06:53,270

 

2094
01:06:53,270 --> 01:06:55,190

 

2095
01:06:55,190 --> 01:06:56,789

 

2096
01:06:56,789 --> 01:06:58,630

 

2097
01:06:58,630 --> 01:07:00,950

 

2098
01:07:00,950 --> 01:07:02,150

 

2099
01:07:02,150 --> 01:07:04,230

 

2100
01:07:04,230 --> 01:07:06,150

 

2101
01:07:06,150 --> 01:07:08,150

 

2102
01:07:08,150 --> 01:07:12,390

 

2103
01:07:12,390 --> 01:07:14,789

 

2104
01:07:14,789 --> 01:07:17,430

 

2105
01:07:17,430 --> 01:07:19,109

 

2106
01:07:19,109 --> 01:07:19,119

 

2107
01:07:19,119 --> 01:07:20,069


2108
01:07:20,069 --> 01:07:23,589

 

2109
01:07:23,589 --> 01:07:25,270

 

2110
01:07:25,270 --> 01:07:29,510

 

2111
01:07:29,510 --> 01:07:29,520

 

2112
01:07:29,520 --> 01:07:30,160


2113
01:07:30,160 --> 01:07:30,170

 

2114
01:07:30,170 --> 01:07:34,230


2115
01:07:34,230 --> 01:07:36,630

 

2116
01:07:36,630 --> 01:07:36,640

 

2117
01:07:36,640 --> 01:07:38,150


2118
01:07:38,150 --> 01:07:40,069

 

2119
01:07:40,069 --> 01:07:42,549

 

2120
01:07:42,549 --> 01:07:44,950

 

2121
01:07:44,950 --> 01:07:46,630

 

2122
01:07:46,630 --> 01:07:48,150

 

2123
01:07:48,150 --> 01:07:48,160

 

2124
01:07:48,160 --> 01:07:49,109


2125
01:07:49,109 --> 01:07:51,349

 

2126
01:07:51,349 --> 01:07:54,789

 

2127
01:07:54,789 --> 01:07:56,630

 

2128
01:07:56,630 --> 01:08:00,230

 

2129
01:08:00,230 --> 01:08:01,910

 

2130
01:08:01,910 --> 01:08:03,190

 

2131
01:08:03,190 --> 01:08:05,109

 

2132
01:08:05,109 --> 01:08:07,029

 

2133
01:08:07,029 --> 01:08:07,039

 

2134
01:08:07,039 --> 01:08:09,109


2135
01:08:09,109 --> 01:08:12,390

 

2136
01:08:12,390 --> 01:08:12,400

 

2137
01:08:12,400 --> 01:08:14,549


2138
01:08:14,549 --> 01:08:16,390

 

2139
01:08:16,390 --> 01:08:17,749

 

2140
01:08:17,749 --> 01:08:19,430

 

2141
01:08:19,430 --> 01:08:20,870

 

2142
01:08:20,870 --> 01:08:23,349

 

2143
01:08:23,349 --> 01:08:23,359

 

2144
01:08:23,359 --> 01:08:25,349


2145
01:08:25,349 --> 01:08:27,910

 

2146
01:08:27,910 --> 01:08:27,920

 

2147
01:08:27,920 --> 01:08:29,990


2148
01:08:29,990 --> 01:08:34,229

 

2149
01:08:34,229 --> 01:08:34,239

 

2150
01:08:34,239 --> 01:08:38,309


2151
01:08:38,309 --> 01:08:40,789

 

2152
01:08:40,789 --> 01:08:42,470

 

2153
01:08:42,470 --> 01:08:43,749

 

2154
01:08:43,749 --> 01:08:48,070

 

2155
01:08:48,070 --> 01:08:50,070

 

2156
01:08:50,070 --> 01:08:50,080

 

2157
01:08:50,080 --> 01:08:51,030


2158
01:08:51,030 --> 01:08:54,149

 

2159
01:08:54,149 --> 01:08:56,149

 

2160
01:08:56,149 --> 01:08:57,829

 

2161
01:08:57,829 --> 01:08:59,110

 

2162
01:08:59,110 --> 01:09:01,669

 

2163
01:09:01,669 --> 01:09:03,590

 

2164
01:09:03,590 --> 01:09:06,149

 

2165
01:09:06,149 --> 01:09:08,309

 

2166
01:09:08,309 --> 01:09:10,070

 

2167
01:09:10,070 --> 01:09:11,510

 

2168
01:09:11,510 --> 01:09:15,430

 

2169
01:09:15,430 --> 01:09:17,189

 

2170
01:09:17,189 --> 01:09:18,829

 

2171
01:09:18,829 --> 01:09:22,630

 

2172
01:09:22,630 --> 01:09:22,640

 

2173
01:09:22,640 --> 01:09:23,590


2174
01:09:23,590 --> 01:09:25,349

 

2175
01:09:25,349 --> 01:09:27,189

 

2176
01:09:27,189 --> 01:09:30,070

 

2177
01:09:30,070 --> 01:09:31,910

 

2178
01:09:31,910 --> 01:09:33,349

 

2179
01:09:33,349 --> 01:09:33,359

 

2180
01:09:33,359 --> 01:09:37,829


2181
01:09:37,829 --> 01:09:37,839

 

2182
01:09:37,839 --> 01:09:44,149


2183
01:09:44,149 --> 01:09:45,430

 

2184
01:09:45,430 --> 01:09:50,390

 

2185
01:09:50,390 --> 01:09:50,400

 

2186
01:09:50,400 --> 01:09:51,410


2187
01:09:51,410 --> 01:09:51,420

 

2188
01:09:51,420 --> 01:09:52,709


2189
01:09:52,709 --> 01:09:54,950

 

2190
01:09:54,950 --> 01:09:56,550

 

2191
01:09:56,550 --> 01:09:56,560

 

2192
01:09:56,560 --> 01:09:59,110


2193
01:09:59,110 --> 01:10:00,709

 

2194
01:10:00,709 --> 01:10:02,790

 

2195
01:10:02,790 --> 01:10:05,110

 

2196
01:10:05,110 --> 01:10:06,630

 

2197
01:10:06,630 --> 01:10:06,640

 

2198
01:10:06,640 --> 01:10:08,070


2199
01:10:08,070 --> 01:10:09,750

 

2200
01:10:09,750 --> 01:10:11,590

 

2201
01:10:11,590 --> 01:10:14,229

 

2202
01:10:14,229 --> 01:10:15,830

 

2203
01:10:15,830 --> 01:10:15,840

 

2204
01:10:15,840 --> 01:10:18,709


2205
01:10:18,709 --> 01:10:21,030

 

2206
01:10:21,030 --> 01:10:23,669

 

2207
01:10:23,669 --> 01:10:26,790

 

2208
01:10:26,790 --> 01:10:28,310

 

2209
01:10:28,310 --> 01:10:29,750

 

2210
01:10:29,750 --> 01:10:31,830

 

2211
01:10:31,830 --> 01:10:34,070

 

2212
01:10:34,070 --> 01:10:34,080

 

2213
01:10:34,080 --> 01:10:34,950


2214
01:10:34,950 --> 01:10:38,790

 

2215
01:10:38,790 --> 01:10:38,800

 

2216
01:10:38,800 --> 01:10:40,709


2217
01:10:40,709 --> 01:10:42,550

 

2218
01:10:42,550 --> 01:10:45,030

 

2219
01:10:45,030 --> 01:10:45,040

 

2220
01:10:45,040 --> 01:10:46,310


2221
01:10:46,310 --> 01:10:48,310

 

2222
01:10:48,310 --> 01:10:49,990

 

2223
01:10:49,990 --> 01:10:51,830

 

2224
01:10:51,830 --> 01:10:53,830

 

2225
01:10:53,830 --> 01:10:55,430

 

2226
01:10:55,430 --> 01:10:57,830

 

2227
01:10:57,830 --> 01:10:59,510

 

2228
01:10:59,510 --> 01:11:01,669

 

2229
01:11:01,669 --> 01:11:01,679

 

2230
01:11:01,679 --> 01:11:02,709


2231
01:11:02,709 --> 01:11:04,950

 

2232
01:11:04,950 --> 01:11:07,669

 

2233
01:11:07,669 --> 01:11:09,590

 

2234
01:11:09,590 --> 01:11:12,229

 

2235
01:11:12,229 --> 01:11:14,310

 

2236
01:11:14,310 --> 01:11:20,149

 

2237
01:11:20,149 --> 01:11:23,270

 

2238
01:11:23,270 --> 01:11:25,270

 

2239
01:11:25,270 --> 01:11:27,430

 

2240
01:11:27,430 --> 01:11:29,669

 

2241
01:11:29,669 --> 01:11:32,070

 

2242
01:11:32,070 --> 01:11:32,080

 

2243
01:11:32,080 --> 01:11:33,189


2244
01:11:33,189 --> 01:11:34,870

 

2245
01:11:34,870 --> 01:11:34,880

 

2246
01:11:34,880 --> 01:11:36,630


2247
01:11:36,630 --> 01:11:38,950

 

2248
01:11:38,950 --> 01:11:41,110

 

2249
01:11:41,110 --> 01:11:44,550

 

2250
01:11:44,550 --> 01:11:47,110

 

2251
01:11:47,110 --> 01:11:53,669

 

2252
01:11:53,669 --> 01:11:54,870

 

2253
01:11:54,870 --> 01:11:57,110

 

2254
01:11:57,110 --> 01:11:59,110

 

2255
01:11:59,110 --> 01:12:00,550

 

2256
01:12:00,550 --> 01:12:01,830

 

2257
01:12:01,830 --> 01:12:03,510

 

2258
01:12:03,510 --> 01:12:03,520

 

2259
01:12:03,520 --> 01:12:04,709


2260
01:12:04,709 --> 01:12:07,270

 

2261
01:12:07,270 --> 01:12:12,630

 

2262
01:12:12,630 --> 01:12:14,390

 

2263
01:12:14,390 --> 01:12:16,870

 

2264
01:12:16,870 --> 01:12:19,590

 

2265
01:12:19,590 --> 01:12:21,110

 

2266
01:12:21,110 --> 01:12:22,790

 

2267
01:12:22,790 --> 01:12:24,229

 

2268
01:12:24,229 --> 01:12:26,550

 

2269
01:12:26,550 --> 01:12:28,229

 

2270
01:12:28,229 --> 01:12:29,669

 

2271
01:12:29,669 --> 01:12:31,350

 

2272
01:12:31,350 --> 01:12:33,270

 

2273
01:12:33,270 --> 01:12:33,280

 

2274
01:12:33,280 --> 01:12:34,310


2275
01:12:34,310 --> 01:12:36,149

 

2276
01:12:36,149 --> 01:12:37,669

 

2277
01:12:37,669 --> 01:12:39,030

 

2278
01:12:39,030 --> 01:12:41,590

 

2279
01:12:41,590 --> 01:12:44,950

 

2280
01:12:44,950 --> 01:12:47,110

 

2281
01:12:47,110 --> 01:12:49,430

 

2282
01:12:49,430 --> 01:12:51,830

 

2283
01:12:51,830 --> 01:12:54,870

 

2284
01:12:54,870 --> 01:12:57,350

 

2285
01:12:57,350 --> 01:12:59,830

 

2286
01:12:59,830 --> 01:13:02,070

 

2287
01:13:02,070 --> 01:13:04,310

 

2288
01:13:04,310 --> 01:13:07,189

 

2289
01:13:07,189 --> 01:13:11,350

 

2290
01:13:11,350 --> 01:13:15,110

 

2291
01:13:15,110 --> 01:13:17,510

 

2292
01:13:17,510 --> 01:13:19,669

 

2293
01:13:19,669 --> 01:13:21,830

 

2294
01:13:21,830 --> 01:13:24,149

 

2295
01:13:24,149 --> 01:13:24,159

 

2296
01:13:24,159 --> 01:13:25,430


2297
01:13:25,430 --> 01:13:27,830

 

2298
01:13:27,830 --> 01:13:29,430

 

2299
01:13:29,430 --> 01:13:31,510

 

2300
01:13:31,510 --> 01:13:32,790

 

2301
01:13:32,790 --> 01:13:34,550

 

2302
01:13:34,550 --> 01:13:35,669

 

2303
01:13:35,669 --> 01:13:37,430

 

2304
01:13:37,430 --> 01:13:40,070

 

2305
01:13:40,070 --> 01:13:42,390

 

2306
01:13:42,390 --> 01:13:42,400

 

2307
01:13:42,400 --> 01:13:43,910


2308
01:13:43,910 --> 01:13:45,270

 

2309
01:13:45,270 --> 01:13:45,280

 

2310
01:13:45,280 --> 01:13:46,390


2311
01:13:46,390 --> 01:13:48,310

 

2312
01:13:48,310 --> 01:13:48,320

 

2313
01:13:48,320 --> 01:13:49,189


2314
01:13:49,189 --> 01:13:50,950

 

2315
01:13:50,950 --> 01:13:50,960

 

2316
01:13:50,960 --> 01:13:51,530


2317
01:13:51,530 --> 01:13:51,540

 

2318
01:13:51,540 --> 01:13:54,229


2319
01:13:54,229 --> 01:13:54,239

 

2320
01:13:54,239 --> 01:13:55,830


2321
01:13:55,830 --> 01:13:57,110

 

2322
01:13:57,110 --> 01:14:01,430

 

2323
01:14:01,430 --> 01:14:01,440

 

2324
01:14:01,440 --> 01:14:02,550


2325
01:14:02,550 --> 01:14:04,630

 

2326
01:14:04,630 --> 01:14:06,630

 

2327
01:14:06,630 --> 01:14:08,470

 

2328
01:14:08,470 --> 01:14:10,870

 

2329
01:14:10,870 --> 01:14:10,880

 

2330
01:14:10,880 --> 01:14:11,990


2331
01:14:11,990 --> 01:14:13,510

 

2332
01:14:13,510 --> 01:14:15,350

 

2333
01:14:15,350 --> 01:14:16,790

 

2334
01:14:16,790 --> 01:14:18,950

 

2335
01:14:18,950 --> 01:14:20,950

 

2336
01:14:20,950 --> 01:14:20,960

 

2337
01:14:20,960 --> 01:14:23,030


2338
01:14:23,030 --> 01:14:26,229

 

2339
01:14:26,229 --> 01:14:28,709

 

2340
01:14:28,709 --> 01:14:30,390

 

2341
01:14:30,390 --> 01:14:31,590

 

2342
01:14:31,590 --> 01:14:33,990

 

2343
01:14:33,990 --> 01:14:34,000

 

2344
01:14:34,000 --> 01:14:34,950


2345
01:14:34,950 --> 01:14:34,960

 

2346
01:14:34,960 --> 01:14:36,070


2347
01:14:36,070 --> 01:14:37,590

 

2348
01:14:37,590 --> 01:14:39,270

 

2349
01:14:39,270 --> 01:14:40,709

 

2350
01:14:40,709 --> 01:14:42,550

 

2351
01:14:42,550 --> 01:14:44,870

 

2352
01:14:44,870 --> 01:14:46,390

 

2353
01:14:46,390 --> 01:14:48,709

 

2354
01:14:48,709 --> 01:14:48,719

 

2355
01:14:48,719 --> 01:14:49,669


2356
01:14:49,669 --> 01:14:52,790

 

2357
01:14:52,790 --> 01:14:57,430

 

2358
01:14:57,430 --> 01:14:59,030

 

2359
01:14:59,030 --> 01:15:00,550

 

2360
01:15:00,550 --> 01:15:00,560

 

2361
01:15:00,560 --> 01:15:03,270


2362
01:15:03,270 --> 01:15:07,590

 

2363
01:15:07,590 --> 01:15:10,070

 

2364
01:15:10,070 --> 01:15:10,080

 

2365
01:15:10,080 --> 01:15:11,270


2366
01:15:11,270 --> 01:15:13,030

 

2367
01:15:13,030 --> 01:15:15,110

 

2368
01:15:15,110 --> 01:15:16,550

 

2369
01:15:16,550 --> 01:15:17,750

 

2370
01:15:17,750 --> 01:15:20,470

 

2371
01:15:20,470 --> 01:15:22,149

 

2372
01:15:22,149 --> 01:15:24,390

 

2373
01:15:24,390 --> 01:15:26,310

 

2374
01:15:26,310 --> 01:15:26,320

 

2375
01:15:26,320 --> 01:15:28,229


2376
01:15:28,229 --> 01:15:31,189

 

2377
01:15:31,189 --> 01:15:33,430

 

2378
01:15:33,430 --> 01:15:35,350

 

2379
01:15:35,350 --> 01:15:37,510

 

2380
01:15:37,510 --> 01:15:39,110

 

2381
01:15:39,110 --> 01:15:40,950

 

2382
01:15:40,950 --> 01:15:42,950

 

2383
01:15:42,950 --> 01:15:47,030

 

2384
01:15:47,030 --> 01:15:49,750

 

2385
01:15:49,750 --> 01:15:51,910

 

2386
01:15:51,910 --> 01:15:53,270

 

2387
01:15:53,270 --> 01:15:55,030

 

2388
01:15:55,030 --> 01:15:57,189

 

2389
01:15:57,189 --> 01:15:57,199

 

2390
01:15:57,199 --> 01:15:58,709


2391
01:15:58,709 --> 01:16:01,510

 

2392
01:16:01,510 --> 01:16:03,189

 

2393
01:16:03,189 --> 01:16:06,550

 

2394
01:16:06,550 --> 01:16:08,550

 

2395
01:16:08,550 --> 01:16:10,870

 

2396
01:16:10,870 --> 01:16:10,880

 

2397
01:16:10,880 --> 01:16:11,830


2398
01:16:11,830 --> 01:16:16,550

 

2399
01:16:16,550 --> 01:16:17,990

 

2400
01:16:17,990 --> 01:16:19,669

 

2401
01:16:19,669 --> 01:16:21,990

 

2402
01:16:21,990 --> 01:16:23,590

 

2403
01:16:23,590 --> 01:16:25,669

 

2404
01:16:25,669 --> 01:16:28,790

 

2405
01:16:28,790 --> 01:16:30,470

 

2406
01:16:30,470 --> 01:16:32,470

 

2407
01:16:32,470 --> 01:16:33,910

 

2408
01:16:33,910 --> 01:16:37,189

 

2409
01:16:37,189 --> 01:16:37,199

 

2410
01:16:37,199 --> 01:16:39,510


2411
01:16:39,510 --> 01:16:41,510

 

2412
01:16:41,510 --> 01:16:43,110

 

2413
01:16:43,110 --> 01:16:46,229

 

2414
01:16:46,229 --> 01:16:47,830

 

2415
01:16:47,830 --> 01:16:48,870

 

2416
01:16:48,870 --> 01:16:50,070

 

2417
01:16:50,070 --> 01:16:51,990

 

2418
01:16:51,990 --> 01:16:52,000

 

2419
01:16:52,000 --> 01:16:53,590


2420
01:16:53,590 --> 01:16:57,430

 

2421
01:16:57,430 --> 01:16:57,440

 

2422
01:16:57,440 --> 01:16:58,709


2423
01:16:58,709 --> 01:17:00,310

 

2424
01:17:00,310 --> 01:17:00,320

 

2425
01:17:00,320 --> 01:17:02,709


2426
01:17:02,709 --> 01:17:05,110

 

2427
01:17:05,110 --> 01:17:05,120

 

2428
01:17:05,120 --> 01:17:08,070


2429
01:17:08,070 --> 01:17:11,430

 

2430
01:17:11,430 --> 01:17:13,910

 

2431
01:17:13,910 --> 01:17:15,910

 

2432
01:17:15,910 --> 01:17:17,990

 

2433
01:17:17,990 --> 01:17:20,070

 

2434
01:17:20,070 --> 01:17:20,080

 

2435
01:17:20,080 --> 01:17:21,750


2436
01:17:21,750 --> 01:17:21,760

 

2437
01:17:21,760 --> 01:17:23,990


2438
01:17:23,990 --> 01:17:25,910

 

2439
01:17:25,910 --> 01:17:28,630

 

2440
01:17:28,630 --> 01:17:30,390

 

2441
01:17:30,390 --> 01:17:30,400

 

2442
01:17:30,400 --> 01:17:33,990


2443
01:17:33,990 --> 01:17:36,550

 

2444
01:17:36,550 --> 01:17:36,560

 

2445
01:17:36,560 --> 01:17:37,430


2446
01:17:37,430 --> 01:17:39,189

 

2447
01:17:39,189 --> 01:17:41,350

 

2448
01:17:41,350 --> 01:17:41,360

 

2449
01:17:41,360 --> 01:17:42,709


2450
01:17:42,709 --> 01:17:44,070

 

2451
01:17:44,070 --> 01:17:45,990

 

2452
01:17:45,990 --> 01:17:46,000

 

2453
01:17:46,000 --> 01:17:46,709


2454
01:17:46,709 --> 01:17:48,790

 

2455
01:17:48,790 --> 01:17:50,550

 

2456
01:17:50,550 --> 01:17:52,390

 

2457
01:17:52,390 --> 01:17:53,990

 

2458
01:17:53,990 --> 01:17:55,750

 

2459
01:17:55,750 --> 01:17:57,830

 

2460
01:17:57,830 --> 01:18:01,430

 

2461
01:18:01,430 --> 01:18:03,590

 

2462
01:18:03,590 --> 01:18:06,470

 

2463
01:18:06,470 --> 01:18:08,470

 

2464
01:18:08,470 --> 01:18:10,229

 

2465
01:18:10,229 --> 01:18:11,990

 

2466
01:18:11,990 --> 01:18:12,000

 

2467
01:18:12,000 --> 01:18:13,910


2468
01:18:13,910 --> 01:18:16,470

 

2469
01:18:16,470 --> 01:18:16,480

 

2470
01:18:16,480 --> 01:18:17,750


2471
01:18:17,750 --> 01:18:20,870

 

2472
01:18:20,870 --> 01:18:20,880

 

2473
01:18:20,880 --> 01:18:21,750


2474
01:18:21,750 --> 01:18:23,030

 

2475
01:18:23,030 --> 01:18:25,110

 

2476
01:18:25,110 --> 01:18:26,709

 

2477
01:18:26,709 --> 01:18:28,390

 

2478
01:18:28,390 --> 01:18:30,630

 

2479
01:18:30,630 --> 01:18:31,910

 

2480
01:18:31,910 --> 01:18:33,510

 

2481
01:18:33,510 --> 01:18:35,750

 

2482
01:18:35,750 --> 01:18:40,149

 

2483
01:18:40,149 --> 01:18:40,159

 

2484
01:18:40,159 --> 01:18:42,149


2485
01:18:42,149 --> 01:18:43,510

 

2486
01:18:43,510 --> 01:18:46,310

 

2487
01:18:46,310 --> 01:18:49,990

 

2488
01:18:49,990 --> 01:18:51,750

 

2489
01:18:51,750 --> 01:18:53,590

 

2490
01:18:53,590 --> 01:18:53,600

 

2491
01:18:53,600 --> 01:18:54,950


2492
01:18:54,950 --> 01:18:58,149

 

2493
01:18:58,149 --> 01:19:01,830

 

2494
01:19:01,830 --> 01:19:01,840

 

2495
01:19:01,840 --> 01:19:03,910


2496
01:19:03,910 --> 01:19:05,110

 

2497
01:19:05,110 --> 01:19:07,510

 

2498
01:19:07,510 --> 01:19:08,709

 

2499
01:19:08,709 --> 01:19:09,910

 

2500
01:19:09,910 --> 01:19:09,920

 

2501
01:19:09,920 --> 01:19:11,110


2502
01:19:11,110 --> 01:19:14,070

 

2503
01:19:14,070 --> 01:19:16,310

 

2504
01:19:16,310 --> 01:19:17,430

 

2505
01:19:17,430 --> 01:19:19,510

 

2506
01:19:19,510 --> 01:19:19,520

 

2507
01:19:19,520 --> 01:19:21,030


2508
01:19:21,030 --> 01:19:24,470

 

2509
01:19:24,470 --> 01:19:26,390

 

2510
01:19:26,390 --> 01:19:29,030

 

2511
01:19:29,030 --> 01:19:31,910

 

2512
01:19:31,910 --> 01:19:34,310

 

2513
01:19:34,310 --> 01:19:34,320

 

2514
01:19:34,320 --> 01:19:35,189


2515
01:19:35,189 --> 01:19:38,070

 

2516
01:19:38,070 --> 01:19:39,910

 

2517
01:19:39,910 --> 01:19:41,510

 

2518
01:19:41,510 --> 01:19:44,630

 

2519
01:19:44,630 --> 01:19:45,830

 

2520
01:19:45,830 --> 01:19:47,750

 

2521
01:19:47,750 --> 01:19:50,630

 

2522
01:19:50,630 --> 01:19:52,470

 

2523
01:19:52,470 --> 01:19:53,830

 

2524
01:19:53,830 --> 01:19:55,750

 

2525
01:19:55,750 --> 01:19:55,760

 

2526
01:19:55,760 --> 01:19:56,630


2527
01:19:56,630 --> 01:19:58,070

 

2528
01:19:58,070 --> 01:19:59,750

 

2529
01:19:59,750 --> 01:20:01,430

 

2530
01:20:01,430 --> 01:20:03,030

 

2531
01:20:03,030 --> 01:20:05,030

 

2532
01:20:05,030 --> 01:20:07,189

 

2533
01:20:07,189 --> 01:20:09,350

 

2534
01:20:09,350 --> 01:20:10,790

 

2535
01:20:10,790 --> 01:20:12,629

 

2536
01:20:12,629 --> 01:20:14,950

 

2537
01:20:14,950 --> 01:20:17,110

 

2538
01:20:17,110 --> 01:20:19,430

 

2539
01:20:19,430 --> 01:20:21,590

 

2540
01:20:21,590 --> 01:20:24,149

 

2541
01:20:24,149 --> 01:20:26,950

 

2542
01:20:26,950 --> 01:20:28,950

 

2543
01:20:28,950 --> 01:20:31,350

 

2544
01:20:31,350 --> 01:20:31,360

 

2545
01:20:31,360 --> 01:20:34,709


2546
01:20:34,709 --> 01:20:36,950

 

2547
01:20:36,950 --> 01:20:38,629

 

2548
01:20:38,629 --> 01:20:40,229

 

2549
01:20:40,229 --> 01:20:41,990

 

2550
01:20:41,990 --> 01:20:43,990

 

2551
01:20:43,990 --> 01:20:45,189

 

2552
01:20:45,189 --> 01:20:47,590

 

2553
01:20:47,590 --> 01:20:47,600

 

2554
01:20:47,600 --> 01:20:48,470


2555
01:20:48,470 --> 01:20:49,750

 

2556
01:20:49,750 --> 01:20:51,669

 

2557
01:20:51,669 --> 01:20:53,270

 

2558
01:20:53,270 --> 01:20:54,870

 

2559
01:20:54,870 --> 01:20:56,550

 

2560
01:20:56,550 --> 01:20:58,310

 

2561
01:20:58,310 --> 01:21:01,270

 

2562
01:21:01,270 --> 01:21:01,280

 

2563
01:21:01,280 --> 01:21:05,270


2564
01:21:05,270 --> 01:21:05,280

 

2565
01:21:05,280 --> 01:21:06,390


2566
01:21:06,390 --> 01:21:07,669

 

2567
01:21:07,669 --> 01:21:10,470

 

2568
01:21:10,470 --> 01:21:10,480

 

2569
01:21:10,480 --> 01:21:14,390


2570
01:21:14,390 --> 01:21:15,910

 

2571
01:21:15,910 --> 01:21:18,070

 

2572
01:21:18,070 --> 01:21:19,590

 

2573
01:21:19,590 --> 01:21:23,110

 

2574
01:21:23,110 --> 01:21:25,270

 

2575
01:21:25,270 --> 01:21:25,280

 

2576
01:21:25,280 --> 01:21:26,550


2577
01:21:26,550 --> 01:21:27,910

 

2578
01:21:27,910 --> 01:21:29,350

 

2579
01:21:29,350 --> 01:21:29,360

 

2580
01:21:29,360 --> 01:21:32,790


2581
01:21:32,790 --> 01:21:35,030

 

2582
01:21:35,030 --> 01:21:36,229

 

2583
01:21:36,229 --> 01:21:36,239

 

2584
01:21:36,239 --> 01:21:37,590


2585
01:21:37,590 --> 01:21:37,600

 

2586
01:21:37,600 --> 01:21:41,189


2587
01:21:41,189 --> 01:21:42,870

 

2588
01:21:42,870 --> 01:21:44,310

 

2589
01:21:44,310 --> 01:21:46,070

 

2590
01:21:46,070 --> 01:21:47,189

 

2591
01:21:47,189 --> 01:21:50,070

 

2592
01:21:50,070 --> 01:21:51,830

 

2593
01:21:51,830 --> 01:21:55,270

 

2594
01:21:55,270 --> 01:21:56,390

 

2595
01:21:56,390 --> 01:21:57,750

 

2596
01:21:57,750 --> 01:21:58,790

 

2597
01:21:58,790 --> 01:22:00,790

 

2598
01:22:00,790 --> 01:22:03,270

 

2599
01:22:03,270 --> 01:22:04,709

 

2600
01:22:04,709 --> 01:22:04,719

 

2601
01:22:04,719 --> 01:22:06,310


2602
01:22:06,310 --> 01:22:06,320

 

2603
01:22:06,320 --> 01:22:10,070


2604
01:22:10,070 --> 01:22:10,080

 

2605
01:22:10,080 --> 01:22:12,950


2606
01:22:12,950 --> 01:22:14,870

 

2607
01:22:14,870 --> 01:22:18,070

 

2608
01:22:18,070 --> 01:22:20,149

 

2609
01:22:20,149 --> 01:22:22,149

 

2610
01:22:22,149 --> 01:22:24,709

 

2611
01:22:24,709 --> 01:22:26,310

 

2612
01:22:26,310 --> 01:22:28,870

 

2613
01:22:28,870 --> 01:22:30,870

 

2614
01:22:30,870 --> 01:22:30,880

 

2615
01:22:30,880 --> 01:22:32,229


2616
01:22:32,229 --> 01:22:34,629

 

2617
01:22:34,629 --> 01:22:34,639

 

2618
01:22:34,639 --> 01:22:35,669


2619
01:22:35,669 --> 01:22:35,679

 

2620
01:22:35,679 --> 01:22:36,470


2621
01:22:36,470 --> 01:22:36,480

 

2622
01:22:36,480 --> 01:22:37,189


2623
01:22:37,189 --> 01:22:40,629

 

2624
01:22:40,629 --> 01:22:43,030

 

2625
01:22:43,030 --> 01:22:45,350

 

2626
01:22:45,350 --> 01:22:45,360

 

2627
01:22:45,360 --> 01:22:46,950


2628
01:22:46,950 --> 01:22:49,030

 

2629
01:22:49,030 --> 01:22:50,950

 

2630
01:22:50,950 --> 01:22:53,910

 

2631
01:22:53,910 --> 01:22:56,390

 

2632
01:22:56,390 --> 01:22:58,470

 

2633
01:22:58,470 --> 01:23:01,030

 

2634
01:23:01,030 --> 01:23:15,350

 

2635
01:23:15,350 --> 01:23:15,360

 

2636
01:23:15,360 --> 01:23:17,440