n ^ (floor)x = (real)n ^ x . in .NET

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n ^ (floor)x = (real)n ^ x .
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6,2 Properties of Floor
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So, the floor of x is defined by connecting it to the conversion from integers to reals in a simple equivalence. The definition of the floor function is an instance of what is called a Galois connection. In general, a Galois connection relates (or connects) two functions by a simple equivalence of the same shape as that above; Galois connections are used to define a complicated function (like the floor function) by mapping its properties into the properties of a simpler function (like the embedding of integers into the reals). This said, it is useful to adopt the mathematical convention of omitting explicit mention of the embedding function and this is what we do from now on.
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6.2 Properties of Floor
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The first time that one encounters a definition like Definition 6.1, it can be difficult to see how it is used. But, it is not as difficult as it may seem. The first thing we can do is to try to identify some special cases that simplify the definition. Two possibilities present themselves immediately; both exploit the fact that the at-most relation is reflexive. The equation is true for all integers n and reals x. Also, [x\ is by definition an integer. So we can instantiate n to [x\. We get The left side that is obtained [xj ^ L*"J is true, and so the right side is also true, That is,
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This tells us that the floor function rounds down. It returns an integer that is at most the given real value. (Note that this is not the same as rounding towards zero. For negative numbers, rounding down rounds away from zero. So the Java realto-integer conversion coincides with the floor function only for positive values.) The second possibility is to instantiate x to n. This is allowed because every integer is a real. Strictly, however, we are instantiating x to the real value obtained by converting n. We get
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n< [n\ = n^n .
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In this case, it is the right side of the equivalence that is true. So we can simplify to Earlier, we determined that [x\ ^x for all real values x. Instantiating x to n, we get
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6: Number Conversion
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Combining the two inequalities, we have derived that, for all integers n,
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[n\=n .
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(Formally, the property of the at-most relation we use is that it is antisymmetric. That is, for all numbers ra and n,m = n exactly when both m^n and n ^ m.) Note that it is not permissible to instantiate n with some real value x. The defining equation is true for all integers n, but a real value is not an integer. A good understanding of the equivalence operator suggests something else we can do with the defining equation: in general, we have
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This is the rule of contraposition. So the contrapositive of the definition of the floor function is, for all integers n and real x,
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But -i(n^m) = m<n. So
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[xj <n = x <n .
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Equally, using that for integers m and n,m<n =
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[xj + 1 ^n = x <n .
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Now we can exploit reflexivity of the at-most relation again. Instantiating n with [xj + 1 and simplifying we deduce: x < LxJ+1 . Recalling that [xj ^x, we have established
[xj ^ x < LxJ + 1 .
In words, [xj is such that [xj is at most x and x is less than lxj + l. Because [xj is an integer, this defines it uniquely. We can express the unicity by a simple equivalence: for all integers m and all reals x,
ra = lxj = m ^ x < m+l .
Recalling the discussion of integer division, we now ask whether the floor function is monotonic. That is, we want to show that
= x^y .
Here we calculate: LxKlyJ { Definition 6.1, x,n := ;y,lxj }
6.3 Indirect Equality
transitivity of
Thus, the floor function is, indeed, monotonic.
6.3 Indirect Equality
Let us now demonstrate how to derive more complicated properties of the floor function, hi the process we introduce an important technique for reasoning with Galois connections called the rule of indirect equality. The following property illustrates the technique:
L*J I = IVx"]
for all x, Suppose we want to establish this property. It is an equality between two floor values; yet the definition of the floor function, Definition 6.1, seems to suggest that we should prove it by proving the two inequalities