By Jones P.R.
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Extra info for Band generated by two semilattices is regular
But we don’t know whether a is equal to '?. Chapter 3 In that case, let’s skip to the if else. ) 't (equal (second-of (cons a (cons '? )) 't) Here are two easy steps. ) 't (equal (car (cdr (cons a (cons '? ) 't (equal (car (cdr (cons a (cons '? )) 't) Obviously, '? is equal to '?. ) 't 't) 't) Indeed. ) 't (equal '? ) 't (equal '? )) 't) Can we rewrite this if question now? Okay. In the if else, we use second-of. That is indeed obvious. ) 't 't) 't) 43 No, but that is ﬁne. We do not need its value after all.
En ) = body where x1 is e1 , . , xn is en . 46 Chapter 4 Yes. Now that we know what “total” means, we can use the Law of Defun for any total function, including all non-recursive functions and many recursive functions. Good question. Our axioms and laws tell us which expressions have equal values. If an expression does not have a value, our axioms and laws do not apply. Functions that are not total are called partial functions. Here is one. 35 36 37 But why does it matter if a function is total?
In frame 28, the orange expression is in a context. In frame 43 of the next chapter, the orange expression is in a focus. A premise is an if question such that a focus can be found in either the if answer or the if else. In frame 28, why is (equal a 't) a premise? Let’s walk through another example using premises and the ﬁnal Law of Dethm. Here is a new dethm. 30 31 32 In frame 28, (equal a 't) is a premise because the focus is in the if answer. That’s a good idea. Is that really a theorem? (dethm jabberwocky† (x) (if (brillig x) (if (slithy x) (equal (mimsy x) 'borogove) (equal (mome x) 'rath)) (if (uﬃsh x) (equal (frumious x) 'bandersnatch) (equal (frabjous x) 'beamish)))) † Thank 20 you, Lewis Carroll (1832-1898).
Band generated by two semilattices is regular by Jones P.R.