By Laurence A. Tepolt

ISBN-10: 083061785X

ISBN-13: 9780830617852

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Notice that the fact that we used the same variable b in both equations is not semantically significant: pattern variables like these have scope only on the right-hand side of their equation. So we could have used b1 in the first equation and b2 in the second, for example. agda, is quite similar: _||_ : B Ñ B tt || b = tt ff || b = b Ñ B Here the situation is dual to that of conjunction. We again just consider the first argument. If it is tt, then this disjunction wins the jackpot: it will come out tt regardless of the value of the second argument.

Since ˜ ˜ tt evaluates to tt (as we can easily confirm with Control-c Control-n), Agda treats the formula we are trying to prove as equivalent to the trivial formula tt ✑ tt. The two formulas are said to be definitionally equal. The formula tt ✑ tt can be proved with refl, and so we complete our definition of ˜˜tt by saying that ˜˜tt is defined to be refl. agda if you are interested. We have proved our first theorem in Agda. Time to celebrate! 3 Going deeper: Curry-Howard and constructivity The Curry-Howard isomorphism was originally developed for constructive logic, and this is still the setting in which it is most used and best known.

In this case, our theorem only has one explicit argument, which is the proof of the assumption. So we just write () there. Agda accepts this definition as a correct proof of our theorem. 2 An alternative proof We could also have proved the theorem this way: ||✑ff1 : ❅ {b1 b2} Ñ b1 || b2 ||✑ff1 {ff} p = refl ||✑ff1 {tt} p = p ✑ ff Ñ b1 ✑ ff This is the same as the proof above, but for the second clause of the definition, we have an actual defining equation which just returns the proof p of the impossible assumption.

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Assembly Language Programming for the TRS-80 Color Computer by Laurence A. Tepolt

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