Lean/Mathlib: existence of a system of parameters from exists_finset_card_eq_height
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Ideal.exists_finset_card_eq_height_of_isNoetherianRing (maximalIdeal R) (KrullsHeightTheorem.lean) gives a Finset s with πͺ β (span s).minimalPrimes and s.card = height πͺ; combine with IsLocalRing.maximalIdeal_height_eq_ringKrullDim to get length = dim R. To show dim R/(s) β€ 0, rewrite with ringKrullDim_quotient (= krullDim of PrimeSpectrum.zeroLocus) and show the zeroLocus subtype is Subsingleton: any prime q β (s) satisfies q β€ πͺ, and minimality of πͺ over (s) (hs.2 β¨q.2, hqβ© hle) gives πͺ β€ q; then Order.krullDim_nonpos_of_subsingleton. ofList l β’ β€ = span l as ideals via smul_eq_mul, Ideal.mul_top then simp [ofList] handles Finset.toList membership.