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b/TypeTheoryAndFormalProofCh3Sec1.typ @@ -51,4 +51,15 @@ $arrow.twohead.r_beta lambda x : (\b\o\o\l->\n\a\t) . space x$ 第二個範䟋就是關斌iteration迭代。 -蚭有型別$delta$和凜敞$F : delta->delta$定矩$D_(delta,F)$是凜敞$lambda x : delta . F(F(x))$。所以$D_(delta,F)$是$F$的第二迭代也暙蚘為$D circle.stroked.small D$。 \ No newline at end of file +蚭有型別$sigma$和凜敞$F : sigma->sigma$定矩$D_(sigma,F)$是凜敞$lambda x : sigma . F(F(x))$。所以$D_(sigma,F)$是$F$的第二迭代也暙蚘為$D circle.stroked.small D$。我們劂䜕讓$sigma$䞍是指單䞀個型別而是任意型別的代皱呢我們又可以甚什麌方法讓$F$可以是任意凜敞呢我們可以甚䞋列衚瀺方匏 + +$D equiv lambda alpha : ast . lambda f : alpha->alpha . space lambda x : a f(f x)$$D$叫做倚型polymorphic凜敞第䞀個抜象abstration是二階second-order的。因歀這暣 + +$D space \n\a\t arrow.twohead.r_(beta) lambda f : \n\a\t -> \n\a\t space . lambda x : \n\a\t space f(f x)$其䞭 $lambda x . f(f x)$即$f circle.stroked.small f$ + +及倖$D space \n\a\t space \s arrow.twohead.r_(beta) lambda s. space s(s x)$這是凜敞將自然敞$x$蜉成$s (s x)$亊即「```function(x){x+2}```」。至斌衚瀺凜敞合成的$x space circle.stroked.small space y$芁劂䜕衚瀺 + +我們可以甚䞋列衚瀺方匏 +$circle.stroked.small equiv lambda alpha:ast. lambda beta : ast. lambda gamma : ast .lambda f : alpha -> beta . lambda g : beta -> gamma . lambda x. g (f x)$ + +å› æ­€$circle.stroked.small space A space B space C space F space G $可以beta-reduction蜉為$G circle.stroked.small F$即$lambda x : A . 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