Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

344 jeremy avigadby (auto simp add: finprod-const prems)finally show ?thesisby (auto simp add: prems)qedThe notation (in comm-group) indicates that this is a lemma in the commutativegroup locale. The notation ( ∧ ) denotes exponentiation, the expression carrier Gdenotes the set underlying the group, and the notation ⊗ x:carrier G denotesthe product over the elements of that set. With this lemma in place, here is aproof of Euler’s theorem:Theorem euler-theorem:assumes m > 1 and zgcd(a, m) = 1shows [a ∧ phi m = 1] (mod m)proof–interpret comm-group [residue-mult-group m]by (rule residue-group-comm-group)have (a modm) { ∧ }m (phi m) ={1}mby (auto simp add: phi-def power-order-eq-one prems)also have (a modm) { ∧ }m (phi m) = (a ∧ phi m) mod mby (rule res-pow-cong)finally show ?thesisby (subst zcong-zmod-eq, insert prems auto simp add: res-one2-eq)qedThe proof begins by noting that the residues modulo m from an abelian group.The notation { ∧ }m then denotes the operation of exponentiation modulom on residues, and {1}m denotes the residue 1, viewed as the identity inthe group. The proof then simply appeals to the preceding lemma, notingthat exponentiating in the group is equivalent to applying ordinary integerexponentiation and then taking the remainder modulo m.The relative simplicity of these proof scripts belies the work that has goneinto making the locales work effectively. The system has to provide mechanismsfor identifying certain theorems as theorems about groups; for specializing factsabout groups, automatically, to the specific group at hand; and for keepingtrack of the calculational rules and basic facts that enable the automated toolsto recognize straightforward inferences and calculations in both the generaland particular settings. Furthermore, one needs mechanisms to manage localesand combine them effectively. For example, every abelian group is a group;the multiplicative units of any ring form a group; the collection of subgroupsof a group has a lattice structure; the real numbers as an ordered field provideinstances of an additive group, a multiplicative group, a field, a linear ordering,and so on. Thus complex bookkeeping is needed to keep track of the facts and