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welcome everybody to this

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episode of functional programming 101

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and this lecture we're going to do
something slightly different

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we're not going to introduce Neil
concepts

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what we're going to do is we're going to
show you

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how 8 temple Kol an

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developments transformational
development

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of programs in Haskell works an

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so the general pattern as that

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you start red and obviously correct
solution

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that usually uses a lot of brute force

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and is cheaper and efficient and then

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usury factoring to make it more
efficient

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right the first goal is a ride is
simplest program

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that works no matter how inefficient it
is

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and then you start re factoring it so

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there's many similarities with you know
extreme programming and things like that

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you know

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just do the men all things to make it
work and then later

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we welcome have tried to optimize it and
maybe it's not necessary to optimize

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because even if something is spread
Reddick maybe you never use it on big &

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Boch

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a or all the inputs that you get can
rerun in reasonable time

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and you can spend your time other on
other things show

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no premature optimization make sure

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that things are correct first and then
optimize

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later month Inc we won't show here

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is that you again also take your hat
school code

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use it as a prototype once you're happy
with it

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you and emblem and the same

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algorithm in another language that this
may be more efficient but

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ya and depending

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on that particular program it mites the

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Haskell Compiler might already good do a
good job

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but sometimes is not just about speech
that you want to use a different

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language

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but it is because

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and in your company at they're not using
haskell but say at Php

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K and remember one of the models of this
course

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is think like a fundamentalist

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but gold like a hacker I'd so

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and this is what we will see here we
start it very fundamentalist saying it

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has to be correct

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and then we gonna be a lil HEC on the
code

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to make it more efficient and abroad the

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problem we're going to look at is a show
called

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Condon problem and come down problem

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is based on a quest program on British
TV that has been there for a while

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as originally based on a French TV show

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an and in that show

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there's a a game that a.m. recall here

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take on time problem and if you

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Google for should go crucial for an ass
call or

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if you read and have Richard birch
papers you will

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see very similar development this is

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a.m. very idiomatic a Haskell
development

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let's look at what this goddamn problem
is first in an informal way

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we need to do is given

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a set of numbers and the standards

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operators addition the getting

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and subtraction and multiplication and
division

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we need to create an expression who is

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value is some given values in this case
we say

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we have too great an expression whose
value is 6

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a 765 now for humans

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this can be hard because the search
space

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as rather bake so we have to go to dry
all possible combinations right brain

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disease

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and whatever to find the answer so this
is fun in a quiz show

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an and some people are really good at it
can do it fast other people

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got bail an but this is something
computers are good at

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right because computers have infinite
patience

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show they can charger dish am
church-based

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and find a solution and if we can
restrict the search page enough

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it might even be able to find the
solutions

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very quickly and that will be the case

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for this example K there's a couple more
rows

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I'm is that all numbers including

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intermediate results must be natural
numbers they must be positive

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each of the numbers in the original Sat

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can be used only once an

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and then there's another show the brawls
but you know

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we keep things simple i right here so

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in particular that should get a possible
solution

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am in our case I'm using the numbers

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25 1050 and long we can create

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7-under 65 as follows: and notice that

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ya 25 -10 is a positive number

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fifty plus one is a positive number and

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when you multiply to positive numbers
they are

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an positive as well alright

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so for this particular example the
search space

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consist of 780 possible solutions

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and but if we should lightly change

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the a result value then

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generally we can find no solution okay

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so we go from 780 possible solutions

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to notion ish okay

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that start to build this a solution to
this program

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bottom-up as they were going to start

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by building the pieces and then we glue
everything together to solve the problem

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up iraq

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on make a mess

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okay every cup so what we're going to do

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sure it will be fine are

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an operators as an algebraic data type
so we have addition

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subtraction multiplication and division
and what we're going to do is we're

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going to define

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a evaluation function that takes to

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numbers and applies

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the operator okay and its

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straightforward whenever you want to add
two numbers

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you add two numbers whenever wherever
you want to ship struck them you should

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check them

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et cetera and the region where using

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this representation and not immediately
evaluating them

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is because we want to build and

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and symbolic representation of the
expression

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an because we we don't want to find the
number

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765 we want to find an expression that
when you evaluate it

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well a.m. return this number

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so we want to do symbolic processing of

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a.m. expressions and then once you have
an expression we can

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evaluated reddish function knowledge and
goat

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and the rules of the game so one of the
rules loss

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that an every operator

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must be applied to positive numbers

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so let's check whether when we

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apply an operator that its FL

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application of that operate show every
at two numbers

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well we don't have to check egos

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if we add to positive numbers the result
will be positive

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every ship tracks two numbers we must
make sure

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that the first number is bigger than the
second number

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because otherwise the result would be
negative and that there's not a lot

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multiplying to positive numbers

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is always positive and when we divide
two numbers

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we have to check that they are actually

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die visible because otherwise we don't
get

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a natural number

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okay and now we can define the data type

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of expressions just ever as we have seen

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be four so we have an slightly different
instead of having like

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have separate operate as a separate
constructor for each operator we have

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Wong application that takes these
operators

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good so how can we evaluate an
expression

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well we evaluate an expression

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and we either return a single din list

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that contains the value or

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when the expression is not valid

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anywhere inside we return the anti
listed

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and you will recognize this trick

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from our partners right are bars is also

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returned either and MD lest when they
failed

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are singled unless to read the results
Monday succeeded

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we don't know we don't use the maybe
type

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were using last because it much more
convenient

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a ride so if you are has school
programmer

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and you're using it may be dyed please
reconsider to use lists

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because as you see here an

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its its little bit and nicer because you
can use

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list comprehension like this okay

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show that see how devaluation works
again

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is defined recursively over expressions

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we either have a value well in that case
it succeeds

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if the expression

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is the value is greater than 0 and every

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have an expression that is the
application oven operator to to ship

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expressions

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what do we do we evaluated the left hand
side

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we've evaluate the right and shiet
recheck

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that the application of this operator
dude East numbers is that let

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and if that's the case be return

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that application there

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gays or really what you see here is dead
engine

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evaluation intermixed with a filter

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for only valid sup expressions I

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to now lets them but they had to
formalize the problem there's a

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a few more help for French is that we
need and these helper functions I will

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just issue that they exist

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but I am you now you will get them

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in your exercises so an Hannah

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I can make sure that they exist but you
will have to define them

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that's the yeah the advantage of
teaching a course

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so the first function here that we're
going to use is Joyce's

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so joyce's takes a list of values of
Taipei

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and readers a list of lists that

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issue all possible ways of choosing zero
or more

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elements from this a.m. input list

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thank it's not as

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complicated Shasha sugar at a concrete
example

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so I want to have all the choices on the
list want to

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these are all the possible ways of
choosing zero or more elements from that

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list so

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I can choose 0 Allemands I could use one
al amanecer that can be one or two

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and then I can choose want to

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Oregon jewish tu wong biggest at

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those are two possible ways to choose to
values

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from that list alright so you see

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that thing here and again I will issue

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that this function exists you will have
to define that

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in the exercises now we can define a new
function

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that given an expression returns

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a list of all the values and

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show all the value show while our dollar
stores are all the values and beliefs

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so if I ever value I returned a single
the last

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if I have the publication of whatever
operator

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I take the value shoved a left

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expression and combine them redevelop
use

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up

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right self-expression so now we can

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decides if some expression is a solution

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00:13:08,870 --> 00:13:12,430
for a given list of numbers by

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more or less directly translating the
rules the game

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00:13:17,160 --> 00:13:20,970
in tunis haskell program show given an
expression

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00:13:20,970 --> 00:13:24,870
this is our list of initial numbers and

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00:13:24,870 --> 00:13:28,810
this is our target number now we have to
check

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whether dish expression here is a
solution

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to the problem okay so what this says is
dead

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first of all the expression must
evaluate

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to the number that were looking for a
show that was a number that we're

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required to find show

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I'm evaluate the expression it must

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and use that number and then

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when we take all the values in that
expression

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everyday call the possible choices

227
00:14:02,050 --> 00:14:05,329
for the input numbers

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00:14:05,329 --> 00:14:08,800
that thing deadliest

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00:14:08,800 --> 00:14:12,110
must be a member of that list

230
00:14:12,110 --> 00:14:15,250
right so if you go back

231
00:14:15,250 --> 00:14:18,699
to the English description of the
comedown problem

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00:14:18,699 --> 00:14:21,870
you will see that this is a very nice

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00:14:21,870 --> 00:14:27,240
direct translation of that definition
nah lets

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00:14:27,240 --> 00:14:31,570
a.m. look at a brute-force solution that
sure church

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00:14:31,570 --> 00:14:34,940
for all a.m. possible solutions

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00:14:34,940 --> 00:14:39,550
and to do that we will use another
function

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00:14:39,550 --> 00:14:42,990
that you will have to be fine in the
exercises

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and ish is a function split and split

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take Celeste and readers a pair of lists

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inside another lest

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00:14:54,470 --> 00:14:58,490
and it will if you all the possible ways

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did you can split deadliest into a pair
of lists

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00:15:02,970 --> 00:15:06,490
roger is on display at 1234

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I get splendid into Wong

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00:15:10,029 --> 00:15:14,149
234 again split and 1234

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00:15:14,149 --> 00:15:14,750
ago

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00:15:14,750 --> 00:15:17,860
split and 123 and for and

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00:15:17,860 --> 00:15:21,040
notes that we don't take into account
the empty list

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00:15:21,040 --> 00:15:24,840
so splitting 1234 en tu wong

250
00:15:24,840 --> 00:15:29,160
and 1234 doesn't really make sense
because then I have not

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00:15:29,160 --> 00:15:34,260
really split the list I'm just gonna
yeah split of the anti list

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00:15:34,260 --> 00:15:37,640
which is gonna nonsensical okay so again

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00:15:37,640 --> 00:15:41,250
this is a function that you will have to
ride in the exercise

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00:15:41,250 --> 00:15:44,350
alright

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00:15:44,350 --> 00:15:48,270
show now began am define a function

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00:15:48,270 --> 00:15:52,780
that given a list of integers well
return

257
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a list expressions whose values that are
in that expression are exist

258
00:15:58,630 --> 00:16:01,920
exactly that list of given numbers

259
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so if we're given an empty list of
numbers

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well then we can build no expression if
we are given

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a single to left well the only
expression we can build

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that contains that value is the
expression

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that on they said Kelly now

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if we are given day lest of numbers

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how can be built all the expressions
well we split the list

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and then recursively use the left side

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too great an expression the right side
too great an expression

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and then we're using are combine
function

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to combine the two right so this is a
brute-force

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way of enumerating

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all possible expressions that are built

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out of these numbers okay and the
combine function here

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just tries to combine these two
expressions

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using any of the operators and I'm what
they can do

275
00:17:12,250 --> 00:17:15,959
as we can right less comprehension that
says

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00:17:15,959 --> 00:17:19,290
if we all the choices for niche an

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00:17:19,290 --> 00:17:23,900
the input list brute force create all
the expressions

278
00:17:23,900 --> 00:17:28,280
and make sure that's the value that i'm
looking for

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00:17:28,280 --> 00:17:31,920
ish the the required value

280
00:17:31,920 --> 00:17:37,240
okay so this

281
00:17:37,240 --> 00:17:40,940
solution here is a more or less

282
00:17:40,940 --> 00:17:44,100
121 translation of the original problem

283
00:17:44,100 --> 00:17:47,670
it's super brute force but it works

284
00:17:47,670 --> 00:17:51,370
okay so it works every run this on

285
00:17:51,370 --> 00:17:55,200
I get I A an very very

286
00:17:55,200 --> 00:17:59,290
am slow laptop an every

287
00:17:59,290 --> 00:18:04,520
look for dissolution storage in a
problem it finds the first solution

288
00:18:04,520 --> 00:18:08,540
and under half a second and under a
minute

289
00:18:08,540 --> 00:18:12,210
it finds all solutions a show even for
dish

290
00:18:12,210 --> 00:18:15,430
very simple abroad for solution

291
00:18:15,430 --> 00:18:18,520
an this is gonna be a fast

292
00:18:18,520 --> 00:18:22,340
an and God takes like an hour about two
minutes to find

293
00:18:22,340 --> 00:18:25,910
all the solutions the question is

294
00:18:25,910 --> 00:18:29,650
can we do better and we definitely can

295
00:18:29,650 --> 00:18:33,920
since we are brute forcing all possible

296
00:18:33,920 --> 00:18:38,150
expressions there will be many
expressions that are invalid

297
00:18:38,150 --> 00:18:41,350
and that we're gonna have to filter out
later show if Megan

298
00:18:41,350 --> 00:18:46,450
make sure that we're not generating any
invalid expressions that my

299
00:18:46,450 --> 00:18:49,810
that it's definitely a win am and

300
00:18:49,810 --> 00:18:53,140
just let's do a back-of-the-envelope
calculation here

301
00:18:53,140 --> 00:18:57,140
there's possible all expressions that we
can build are

302
00:18:57,140 --> 00:19:01,350
33 million and only 5 a million of them

303
00:19:01,350 --> 00:19:05,360
are actually sell it K and then

304
00:19:05,360 --> 00:19:08,380
what we will do is we will combine

305
00:19:08,380 --> 00:19:12,580
the generation function with the
evaluation function so we will fuse the

306
00:19:12,580 --> 00:19:13,250
SDO

307
00:19:13,250 --> 00:19:16,810
to make sure that things are rejected

308
00:19:16,810 --> 00:19:20,480
earlier but before we do that this

309
00:19:20,480 --> 00:19:23,630
sequence loss rather long

310
00:19:23,630 --> 00:19:27,750
show let's take a well-deserved break
and then see you

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00:19:27,750 --> 00:19:29,230
in a bit thank you