number-theory

6B The Curry-Howard isomorphism 83term-variables).a-*a--*ca-+ca(-E)a(-+E)c(ii) If A is the logic deduction in 6A2.2, AA is the TA2-deduction in 6B1.1 (modulo=a)(iii) If A is the logic deduction in 6A3.1, AA is the TA2-deduction in 6B1.2 (modulo=a)6B4.3 Exercise* Construct and compare (A5)2 and (A6)2, where A5 is the deductionin 6A3.3 and A6 is the following deduction.[a] (00)a-+a (0)(-'I)(-+I)aa vacouously}6B5 Logic-to-Lambda Lemma Let al_., an H i by a logic deduction A, and foreach i < n letbe those occurrences of 61 that are undischarged assumptions at branch-tops in A. ThenAA is defined and is a TA2-deduction whose conclusion has formx1 t:oli...,x1mi:v1, ... xn,l:o ...... :an '- M:2where each x1j occurs exactly once in M (at the same position as q.J has in A) andM has no bound-variable clashes. Also(AA)L = A.Proof Straightforward induction on the number of steps in A. The change ofvariables in 6B4(ii) ensures that M has no clashes of bound variables and has onlyone occurrence of each of its free variables.6135.1 Note The Curry-Howard mapping from A to AL is usually called the Curry-Howard isomorphism: how far can it justify this description? There are three levelsto consider:(i) provable formulae H types of closed terms,(ii) logic proofs -> TAx-proofs,(iii) logic deductions +- TA1-deductions.By 6B5, the AL-mapping has a one-sided inverse AA such that (A2)L A, but for itto be an isomorphism in any real sense we should also have (AL)z A. We haveseen in 6B3 that this breaks down on level (iii). But on levels (i) and (ii) it holdstrue, as the next theorem will show.