66 lines
2 KiB
Text
66 lines
2 KiB
Text
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.SMNu.PlusU1.Basic
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import HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps
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import HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols
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universe v u
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namespace SMRHN
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namespace PlusU1
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open SMνCharges
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open SMνACCs
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open BigOperators
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def existsPlane (n : ℕ) : Prop := ∃ (B : Fin n → (PlusU1 3).charges),
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LinearIndependent ℚ B ∧ ∀ (f : Fin n → ℚ), (PlusU1 3).isSolution (∑ i, f i • B i)
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lemma exists_plane_exists_basis {n : ℕ} (hE : existsPlane n) :
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∃ (B : Fin 11 ⊕ Fin n → (PlusU1 3).charges), LinearIndependent ℚ B := by
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obtain ⟨E, hE1, hE2⟩ := hE
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obtain ⟨B, hB1, hB2⟩ := eleven_dim_plane_of_no_sols_exists
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let Y := Sum.elim B E
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use Y
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apply Fintype.linearIndependent_iff.mpr
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intro g hg
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rw [@Fintype.sum_sum_type] at hg
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rw [@add_eq_zero_iff_eq_neg] at hg
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rw [← @Finset.sum_neg_distrib] at hg
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have h1 : ∑ x : Fin n, -(g (Sum.inr x) • Y (Sum.inr x)) =
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∑ x : Fin n, (-g (Sum.inr x)) • Y (Sum.inr x):= by
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apply Finset.sum_congr
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simp
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intro i _
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simp
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rw [h1] at hg
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have h2 : ∑ a₁ : Fin 11, g (Sum.inl a₁) • Y (Sum.inl a₁) = 0 := by
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apply hB2
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erw [hg]
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exact hE2 fun i => -g (Sum.inr i)
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rw [Fintype.linearIndependent_iff] at hB1 hE1
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have h3 : ∀ i, g (Sum.inl i) = 0 := hB1 (fun i => (g (Sum.inl i))) h2
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rw [h2] at hg
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have h4 : ∀ i, - g (Sum.inr i) = 0 := hE1 (fun i => (- g (Sum.inr i))) hg.symm
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simp at h4
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intro i
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match i with
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| Sum.inl i => exact h3 i
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| Sum.inr i => exact h4 i
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theorem plane_exists_dim_le_7 {n : ℕ} (hn : existsPlane n) : n ≤ 7 := by
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obtain ⟨B, hB⟩ := exists_plane_exists_basis hn
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have h1 := LinearIndependent.fintype_card_le_finrank hB
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simp at h1
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rw [show FiniteDimensional.finrank ℚ (PlusU1 3).charges = 18 from
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FiniteDimensional.finrank_fin_fun ℚ] at h1
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exact Nat.le_of_add_le_add_left h1
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end PlusU1
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end SMRHN
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