Loop Invariants - Principles of Imperative Computation (Carnegie Mellon University)

00:10:50
https://www.youtube.com/watch?v=7_kiWYYLngQ

摘要

TLDRIn this lecture, the speaker discusses contracts in programming, focusing on assertions, preconditions, postconditions, and loop invariants. Assertions are defined as true or false statements that validate conditions necessary for functions to operate correctly. The importance of ensuring loops terminate and produce the desired output is emphasized, introducing loop invariants as Boolean statements that must hold true at the start and end of loop iterations. The lecture provides examples to illustrate how to identify and prove loop invariants, as well as how to demonstrate loop termination and derive postconditions from these invariants.

心得

  • 🔍 Assertions are true/false statements.
  • 📋 Preconditions ensure function validity.
  • ✅ Postconditions confirm outcomes after execution.
  • 🔄 Loops must terminate and yield results.
  • 📏 Loop invariants are true at loop start/end.
  • 🛠️ Proving loop invariants is crucial for correctness.
  • ⏳ Loop termination prevents infinite loops.
  • 📈 J and K increase in the loop, ensuring termination.
  • 🔗 Postconditions relate to loop invariants.
  • 💡 Understanding these concepts is key in programming.

时间轴

  • 00:00:00 - 00:05:00

    The lecture introduces the concept of contracts in programming, focusing on assertions, preconditions, and postconditions. An assertion is defined as a Boolean statement that can be true or false, with examples illustrating preconditions that must be met for functions to avoid undesirable outcomes. The importance of loops in programming is highlighted, emphasizing the need for loops to terminate and produce the desired output, which leads to the concept of loop invariants—statements that remain true at the start and end of each loop iteration, aiding in proving the correctness of loops.

  • 00:05:00 - 00:10:50

    The discussion continues with examples of loop invariants, demonstrating how to identify them through Boolean statements. The lecture explains the process of analyzing a piece of code to derive loop invariants and prove their validity. It also covers the importance of proving loop termination by showing that certain variables will eventually exceed specified limits. Finally, the relationship between loop invariants and postconditions is explored, concluding with a discussion on how to derive postconditions from loop invariants, setting the stage for further class discussions.

思维导图

视频问答

  • What is an assertion?

    An assertion is a statement that is either true or false, used to validate conditions in programming.

  • What are preconditions and postconditions?

    Preconditions are conditions that must be true before a function executes, while postconditions are conditions that must be true after the function has executed.

  • What is a loop invariant?

    A loop invariant is a Boolean statement that remains true at the start and end of each iteration of a loop.

  • Why are loop invariants important?

    Loop invariants help prove the properties of loops, ensuring their correctness and desired outcomes.

  • How do you prove a loop invariant?

    To prove a loop invariant, you must show that it holds true at the beginning of the loop and remains true after each iteration.

  • What is the significance of loop termination?

    Loop termination ensures that a loop will eventually stop executing, preventing infinite loops.

  • How can postconditions be derived from loop invariants?

    Postconditions can be derived by analyzing the loop invariants and the conditions under which the loop terminates.

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  • 00:00:00
    so in this lecture in this short video
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    lecture I'm going to talk about
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    contracts and uh contracts um I'm going
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    to start with assertions pre and post
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    conditions now what's an assertion an
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    assertion is a statement that's either
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    true or false so it's a Boolean
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    expression that we are trying to assert
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    like for example you can say I want to
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    try to assert that X is greater than or
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    equal to zero right here so we may
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    require that X to be greater than equal
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    Z and as an assertion here a
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    precondition for a function for example
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    we may say that we require at this point
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    the Y to be greater than equal Zer so we
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    make sure that the precondition is met
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    so that function will not have
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    undesirable outcomes the post condition
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    means that we
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    ensures that some condition has come
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    true at the end of this function so we
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    there are three things that are
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    important ass
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    preconditions and post
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    conditions now Loops are really
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    important things in programming we write
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    a loop to iterate over a set of
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    statements again and again and again now
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    one of the things we need to assure that
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    the loop will terminate and also the
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    loop will produce the desired output
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    these two things are really really
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    important now so in order to get there
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    we will start with a concept called a
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    loop invariant so what is is a loop
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    invariant a loop invariant is a Boolean
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    statement that is true at the start of
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    the loop that's true one time in the
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    beginning and then at the end of each
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    iteration this means that the loop
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    invariant must be true immediately after
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    you exit the loop so why do we study
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    Loop invariance we study Loop invariance
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    to prove properties of loops and that
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    will allow us to prove the partial
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    correctness of loops and what the loops
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    will produce what the outcome is and
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    here's an example
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    so let's take this simple uh loop as an
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    example let's try to Define some Boolean
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    statements that may or may not be true
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    as a loop invariant so we can take a
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    very trivial one says 1 equals 1 which
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    is a loop invariant because it never
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    changes inside the loop so that's great
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    but it is kind of
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    useless if you take some statement like
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    I equals I which is also true inside the
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    loop because I is part of the loop but
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    again it's a trivial statement therefore
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    although it is a loop invariant but
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    they're not really
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    useful if you take a statement like I is
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    strictly less than n is that a loop
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    invariant let's try to see that we know
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    that when we start I is equal to zero so
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    if we assume that n is strictly greater
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    than zero to begin with this condition
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    is true what about after one iteration
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    so I prime is I + 1 can we prove that I
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    + 1 is less than strictly less than n
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    the answer is no therefore this is not a
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    loop invariant what if I say I is less
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    than or equal to n well I should be able
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    to prove this now because in the
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    beginning I is z if I assume n is
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    greater than equal to Z as a
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    precondition then I know this statement
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    to be true and the I prime which is
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    equal to I + 1 is now less than or equal
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    to n since
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    I is strictly less than n so we did find
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    a loop invariant called I less than
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    equal to n now if I take the loop
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    invariant I is less than or equal to n
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    which I know is true here outside of the
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    loop and also the loop exit exit
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    condition which says I is greater than
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    or equal to n i can conclude that at the
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    end of this loop I must be equal to
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    n now let's how do we find a loop in
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    variance now this is a process that may
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    take a while so let's think about this
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    piece of cord now we do not we do not
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    know what this does it's some some piece
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    of cod that is doing something now
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    obviously we can use something like coin
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    to figure out given X we can find maybe
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    we can do a few Explorations what is g
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    of Z what is g of one what is g of two
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    and so on we can do that but that
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    doesn't give us a looping variant that
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    may give us a hint about what this
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    function produces so to find a loop
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    invariant we have to come up with a
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    statement a Boolean statement that get
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    involved get involves IGN K perhaps to
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    come up with a statement that's that we
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    can prove so there are four things in
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    this piece of cord x i j and
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    k now we do know that X is a loop
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    invariant because X never changes inside
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    the loop let's start with X to be zero
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    and we do know that initially I and K
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    are all zeros now when you go into the
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    loop the J changes to J + k + 3 I + 1
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    since J K and I are zero the J will
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    become one in the first iteration the K
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    will become k + 6 I + 3 so since K and I
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    are zero and to begin with K would
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    become
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    three and I would be incremented to
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    1 now in the second iteration of the
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    loop the J becomes J + K so we will take
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    1 + 3 that would be 4 3 * I that's 7 + 1
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    so J would become 8 and K would become k
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    + 6 I 3 + 6 I that's 9 + 3 K would
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    become 12 and the I would become two at
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    this point the J is greater than x which
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    is
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    it's a loop invariant then I must prove
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    that this is true in the beginning
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    obviously it is true because I and J and
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    K are all three are zeros so beginning
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    one is okay but what about after one
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    iteration what about I prime Cube now
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    the I prime is the same as I + 1 because
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    I changes to i1 and I + 1 Cub is equal
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    to I Cub + 3 i^ 2 + 3 I +
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    1 and let's look at J Prime now the J
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    Prime is equal to uh J + k + 3 I + 1 so
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    the question we have is can we show that
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    the I
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    prime Cub is equal to J Prime uh well at
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    least I have two terms here 3 I = 3 I
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    and then 1 = 1 but I'm not really sure
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    then I also know this I know that this
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    is true but so if I can prove that K = 3
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    i^ 2 I will be able to prove this so
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    let's try to prove K = 3 i² 3 i s as a
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    loop in variant so to prove that I need
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    to prove that K Prime is equal to 3 * I
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    Prime 2 now what is K Prime K Prime is
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    equal to k + 6 I +
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    3 and I prime is I + 1
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    so if I do this then I'll get 3 i^ 2 +
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    uh 6 I + 3 on this side since K = 3 i^
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    2 I will get the same thing here so
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    hence I can prove that therefore that I
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    prove that this is a looping variant now
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    both of these are looping variants so I
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    can confirm that they are Loop
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    invariance
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    indeed now so the other problem that we
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    have is proving the loop termination now
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    in this case the chode if you look at
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    the chode that the J eventually because
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    J is an increasing function must
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    eventually exceed the X and since J is
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    always increasing at least by one k is
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    always increasing at least by three and
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    I is always increasing at least by one
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    so we do know that J is going to be
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    sometime going to go over X so in order
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    to prove the loop termination we say
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    that
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    since uh J
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    increases okay and the J is going to be
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    at some point greater than or equal to X
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    and the loop will terminate at that
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    point so how do we get the post
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    condition from the loop inv variant now
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    Loop invariant post condition now we
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    have two Loop invariances in this case I
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    cubal J and K = 3 I 2 now it seems to me
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    that the loop condition was that while
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    the J is less than x so it seems to me
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    that since J is equal to I Cub we are
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    testing to find the I whose cube is less
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    than
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    x and so for example when it's four one
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    was the I that we got when it it's two
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    perhaps uh if x was uh something like
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    nine we might be able to get that the uh
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    2 CU which is 8 is less than
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    9 so this is a little bit of uh we'll
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    leave that discussion for class and uh
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    we will end this uh video lecture now
标签
  • contracts
  • assertions
  • preconditions
  • postconditions
  • loop invariants
  • programming
  • correctness
  • termination
  • Boolean statements
  • functions