我喜欢这个非常简单的定义:

循环不变量是(程序变量之间的)一种条件，它必须在每次循环迭代之前和之后立即为真。(请注意，在迭代过程中，这并没有说明它的真伪。)

By itself, a loop invariant doesn't do much. However, given an appropriate invariant, it can be used to help prove the correctness of an algorithm. The simple example in CLRS probably has to do with sorting. For example, let your loop invariant be something like, at the start of the loop, the first i entries of this array are sorted. If you can prove that this is indeed a loop invariant (i.e. that it holds before and after every loop iteration), you can use this to prove the correctness of a sorting algorithm: at the termination of the loop, the loop invariant is still satisfied, and the counter i is the length of the array. Therefore, the first i entries are sorted means the entire array is sorted.

一个更简单的例子:循环不变量、正确性和程序推导。

我理解循环不变量的方式是作为一个系统的，正式的工具来推理程序。我们做了一个陈述，我们专注于证明它是正确的，我们称之为循环不变量。这组织了我们的逻辑。虽然我们也可以非正式地讨论一些算法的正确性，但使用循环不变量迫使我们非常仔细地思考，并确保我们的推理无懈可击。

简单地说，循环不变量是对循环的每次迭代都成立的某个谓词(条件)。例如，让我们看一个简单的For循环，它是这样的:

int j = 9;
for(int i=0; i<10; i++)  
  j--;

在这个例子中，i + j == 9(对于每个迭代)是正确的。一个较弱的不变式也是成立的 I >= 0 && I <= 10。

《如何思考算法》的定义，Jeff Edmonds著

循环不变式是放置在循环和循环顶部的断言 每次计算返回到循环的顶部时，这必须成立。

对不起，我没有评论权限。

正如你提到的@Tomas Petricek

另一个较弱的不变式也是成立的，即i >= 0 && i < 10(因为这是连续条件!)”

为什么它是循环不变量?

我希望我没有错，据我理解[1]，循环不变将在循环开始时为真(初始化)，它将在每次迭代(维护)之前和之后为真，它也将在循环结束后为真(终止)。但是在最后一次迭代之后，i变成了10。因此，条件i >= 0 && i < 10变为假值并终止循环。它违反了循环不变量的第三个性质(终止)。

[1] http://www.win.tue.nl/~kbuchin/teaching/JBP030/notebooks/loop-invariants.html

除了这些不错的答案，我想Jeff Edmonds在《如何思考算法》(How to Think About Algorithms)中举的一个很好的例子可以很好地说明这个概念:

EXAMPLE 1.2.1 "The Find-Max Two-Finger Algorithm" 1) Specifications: An input instance consists of a list L(1..n) of elements. The output consists of an index i such that L(i) has maximum value. If there are multiple entries with this same value, then any one of them is returned. 2) Basic Steps: You decide on the two-finger method. Your right finger runs down the list. 3) Measure of Progress: The measure of progress is how far along the list your right finger is. 4) The Loop Invariant: The loop invariant states that your left finger points to one of the largest entries encountered so far by your right finger. 5) Main Steps: Each iteration, you move your right finger down one entry in the list. If your right finger is now pointing at an entry that is larger then the left finger’s entry, then move your left finger to be with your right finger. 6) Make Progress: You make progress because your right finger moves one entry. 7) Maintain Loop Invariant: You know that the loop invariant has been maintained as follows. For each step, the new left finger element is Max(old left finger element, new element). By the loop invariant, this is Max(Max(shorter list), new element). Mathe- matically, this is Max(longer list). 8) Establishing the Loop Invariant: You initially establish the loop invariant by point- ing both fingers to the first element. 9) Exit Condition: You are done when your right finger has finished traversing the list. 10) Ending: In the end, we know the problem is solved as follows. By the exit condi- tion, your right finger has encountered all of the entries. By the loop invariant, your left finger points at the maximum of these. Return this entry. 11) Termination and Running Time: The time required is some constant times the length of the list. 12) Special Cases: Check what happens when there are multiple entries with the same value or when n = 0 or n = 1. 13) Coding and Implementation Details: ... 14) Formal Proof: The correctness of the algorithm follows from the above steps.

在这种情况下，不变量意味着在每次循环迭代的某一点上必须为真条件。

在契约编程中，不变量是在调用任何公共方法之前和之后必须为真(通过契约)的条件。