generators_minimal

formal source

  64/-! ## Closure properties. -/
  65
  66/-- The set {2, 3, 5} is the smallest generating set for the spectrum. -/
  67theorem generators_minimal :
  68    G2 ≠ G3 ∧ G3 ≠ G5 ∧ G2 ≠ G5 := by
  69  refine ⟨?_, ?_, ?_⟩ <;> decide
  70
  71/-- The product of all three generators: 2 · 3 · 5 = 30 (= primorial p₃#). -/
  72theorem primorial_product : G2 * G3 * G5 = 30 := by decide
  73
  74/-- 30 is the third primorial. The primorial spectrum is RS-canonical:
  75    p₁# = 2, p₂# = 6, p₃# = 30, p₄# = 210. -/
  76theorem second_primorial : G2 * G3 = 6 := by decide
  77theorem third_primorial : G2 * G3 * G5 = 30 := by decide
  78
  79structure RecognitionGeneratorsCert where
  80  generator_2 : G2 = 2
  81  generator_3 : G3 = 3
  82  generator_5 : G5 = 5
  83  primorial_3 : G2 * G3 * G5 = 30
  84  generators_distinct : G2 ≠ G3 ∧ G3 ≠ G5 ∧ G2 ≠ G5
  85  -- Spectrum decompositions
  86  four_as_G : (4 : ℕ) = G2^2
  87  six_as_G : (6 : ℕ) = G2 * G3
  88  fortyfive_as_G : (45 : ℕ) = G3^2 * G5
  89  oneTwentyFive_as_G : (125 : ℕ) = G5^3
  90  threeSixty_as_G : (360 : ℕ) = G2^3 * G3^2 * G5
  91
  92def recognitionGeneratorsCert : RecognitionGeneratorsCert where
  93  generator_2 := rfl
  94  generator_3 := rfl
  95  generator_5 := rfl
  96  primorial_3 := primorial_product
  97  generators_distinct := generators_minimal