Geodesic
Recursive $p$-adic numbers and elementary theories of finitely generated pro-$p$-groups
Izvestiya. Mathematics , Tome 30 (1988) no. 3, pp. 577-597.

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The authors propose a method of effective presentation of finitely generated pro-$p$-groups, and use it to study the elementary theories of such groups. They prove that elementarily equivalent finitely generated pro-$p$-groups are isomorphic. The main result is the following criterion: the elementary theory of a finitely generated nilpotent pro-$p$-group $G$ is decidable if and only if $G$ is effectively presented. Bibliography: 18 titles. 
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A. G. Myasnikov; V. N. Remeslennikov. Recursive $p$-adic numbers and elementary theories of finitely generated pro-$p$-groups. Izvestiya. Mathematics , Tome 30 (1988) no. 3, pp. 577-597. http://geodesic.mathdoc.fr/item/IM2_1988_30_3_a6/

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