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Optimize FiveEighths cardinality bridge #650

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Merged
Vilin97 merged 1 commit into from
Oct 13, 2026
Merged

Optimize FiveEighths cardinality bridge #650

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7 changes: 2 additions & 5 deletions LeanPool/FiveEighthsTheorem/Basic.lean
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Expand Up @@ -132,11 +132,8 @@ private theorem ncard_le_ncard_univ {α : Type*} [Finite α] (p : α → Prop) :
/-- The commuting probability of `G` counts the commuting pairs of elements. -/
theorem commProb_eq_ncard_div_sq (G : Type*) [Group G] :
commProb G = (Set.ncard { (g, h) : G × G | g * h = h * g } : ℚ) / (Nat.card G : ℚ) ^ 2 := by
have key : Nat.card { p : G × G // Commute p.1 p.2 }
= Set.ncard { (g, h) : G × G | g * h = h * g } := by
rw [← Nat.card_coe_set_eq]
exact Nat.card_congr (Equiv.subtypeEquivRight fun p => Iff.rfl)
rw [commProb_def, key]
rw [commProb_def]
congr 1

/-- The cardinality of a type, as a rational number (zero if the type is infinite). -/
noncomputable abbrev typeQCard (G : Type*) : ℚ := Nat.card G
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