Formatting Calculations in .NET

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3.3 Formatting Calculations
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The relation between the original left and right sides is the same as the relation between the left and right sides obtained by carrying out steps 1-4. D
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3.3 Formatting Calculations
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The message of Section 3.2 is that reducing problem solving to calculation can be very effective, but, the use of a verificational, as opposed to constructive, style of reasoning can make simple calculations opaque and complex. In this text, we will develop a style of calculation that aims to elucidate the process of constructing solutions to non-trivial mathematical and programming problems. Throughout, we use a uniform format for presenting calculations. We summarize the format in this section, but discuss it again when we begin to use it in earnest. The reader may therefore wish to omit this section on a first reading, returning to it when the need arises.
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3.3.1 Basic Structure
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Our calculations have a number of steps (usually more than one). A mandatory element is that each step is accompanied by a hint providing a justification for the validity of the step. For example, a two-step calculation might have the following shape.
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{ T .
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In this calculation, R, S and T are expressions, and p and q are hints why R = S and S = T, respectively. The conclusion of the calculation is that R = T. Here is a concrete example, where we use the laws of arithmetic to simplify an arithmetic expression. The goal of the calculation is to simplify the expression (n+1) 2 - n2 by eliminating the squaring operator. (n+1) 2 - n2 { x2-y2 = (x-y)x(x+y),
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withx,y := n + 1 , n } ((n+1) -n) x ((n+1) +n) { addition is symmetric and associative } ((n-n) +1) x ((n+n) + 1)
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3: Calculational Proof
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2xn + 1 .
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n-n = 0, n+n = 2xn }
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( 0 + l ) x ( 2 x n + 1) arithmetic }
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The calculation is parametrized by the variable n which, by convention, denotes an arbitrary (natural) number. The conclusion of the calculation is the equality between the first and last lines: (n+l)2-n2 = 2 x n + l for all (natural) numbers n.
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3.3.2 Hints
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The hints in a calculation serve a number of purposes. The simple device of bracketing allows them to be of any size whatsoever. As a consequence, we may not only give detailed information about the formal justification for an individual step but also, whenever necessary, explain where we are going and why. We can also include a subcalculation in the hint, should we choose to do so. In the above calculation, the hints get progressively simpler so let us begin with the last one. The final hint, 'arithmetic', says almost nothing; it simply says that some property of arithmetic is being used. Here, the general laws actually being used are that 0+x = x, and 1 xx = x, irrespective of the value of x. These are very basic laws, which we expect the reader to be completely familiar with, and the presence of a hint is deemed to be superfluous. In more complicated cases, a hint like 'arithmetic' can be very useful. If, for example, an expression involves both arithmetic and boolean operators, such a hint focuses attention on the arithmetic operators rather than the boolean operators. In contrast, the first hint is quite complicated. It states a property of arithmetic that holds for all values of the variables x and y. The accompanying text 'with x ,y := n+1 , n' indicates that the property is being used in the case that x has the value n+1 and y has the value n. Formally, the equality between the first and second expressions in the calculation (n+l)2-n2 = is the instance of the law
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x2 - y2 = (x-y) x (x+y)
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obtained by instantiating x to n+1 and y to n. Note that n+1 is parenthesized in order to make it clear how the instance is obtained. Sometimes, as in this case, parentheses are unnecessary when forming
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