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User:Michiexile/MATH198/Lecture 8
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====Lambek's lemma==== What we do when we write a recursive data type definition in Haskell ''really'' to some extent is to define a data type as the initial algebra of the corresponding functor. This intuitive equivalence is vindicated by the following '''Lemma''' ''Lambek'' If <math>P: C\to C</math> has an initial algebra <math>I</math>, then <math>P(I) = I</math>. '''Proof''' Let <math>a: PA\to A</math> be an initial <math>P</math>-algebra. We can apply <math>P</math> again, and get a chain :<math>PPA \to^{Pa} PA \to^a A</math> We can fill out the diagram :[[Image:LambekPartialDiagram.png]] to form the diagram :[[Image:LambekDiagram.png]] where <math>f</math> is induced by initiality, since <math>Pa \colon PPA \to PA</math> is also a <math>P</math>-algebra. The diagram above commutes, and thus <math>af = 1_{PA}</math> and <math>fa = 1_A</math>. Thus <math>f</math> is an inverse to <math>a</math>. QED. Thus, by Lambek's lemma we ''know'' that if <math>P_A(X) = 1 + A\times X</math> then for that <math>P_A</math>, the initial algebra - should it exist - will fulfill<math>I = 1 + A\times I</math>, which in turn is exactly what we write, defining this, in Haskell code: <haskell> List a = Nil | Cons a List </haskell>
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