Golod-Shafarevich infinitude criterion for pro-p groups

If a pro-$p$ presentation on $d$ generators and possibly infinitely many relators $r_i$ satisfieswith the sum convergent and $D$ the Magnus degree of a free pro-p group element, then the presented pro-p group is infinite. The Golod-Shafarevich inequality forces its completed group algebra to be infinite-dimensional, which would be impossible for a finite group. This formulation applies to relators whose degrees tend to infinity and is suitable for enumerated power-relation constructions.