Getting tame intersections in Alperin's fusion theorem: a two-case trick after conjugating D

Tameness means N_P(P∩Q) and N_Q(P∩Q) are both Sylow in N_G(P∩Q). In the downward induction on |P ∩ R| with D = P ∩ R and h ∈ N_G(D), pick a Sylow S ⊇ N_P(D) with S ∩ N_G(D) Sylow in N_G(D), and u with S^u = P. Then N_P(D^u) = (S ∩ N(D))^u is Sylow in N(D^u). Put h' = u⁻¹hu ∈ N(D^u) and R₂ = h'Ph'⁻¹, so R₂^{h'} = P and D^u ≤ P ∩ R₂. Case |P ∩ R₂| > |D|: the induction hypothesis handles conjugation by h'. Otherwise P ∩ R₂ = D^u (Subgroup.eq_of_le_of_card_ge), and N_{R₂}(D^u) = N_P(D^u)^{h'⁻¹} is Sylow too, so the move is a tame generator. Chain: x ~ x^u (IH for S), x^u ~ (x^h)^u, (x^h)^u ~ x^h (IH for S again, symmetric). In Lean, encode 'Sylow in N(D)' as 'card ≥ every p-subgroup of N(D)', which transports easily under MulAut.conj maps.