feat: Add results about solution planes

HepLean/AnomalyCancellation/SMNu/PlusU1/BoundPlaneDim.lean

@@ -0,0 +1,65 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.AnomalyCancellation.SMNu.PlusU1.Basic
+import HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps
+import HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols
+
+universe v u
+
+namespace SMRHN
+namespace PlusU1
+
+open SMνCharges
+open SMνACCs
+open BigOperators
+
+def existsPlane (n : ℕ) : Prop := ∃ (B : Fin n → (PlusU1 3).charges),
+   LinearIndependent ℚ B ∧ ∀ (f : Fin n → ℚ), (PlusU1 3).isSolution (∑ i, f i • B i)
+
+lemma exists_plane_exists_basis {n : ℕ} (hE : existsPlane n) :
+    ∃ (B : Fin 11 ⊕ Fin n → (PlusU1 3).charges), LinearIndependent ℚ B := by
+  obtain ⟨E, hE1, hE2⟩ := hE
+  obtain ⟨B, hB1, hB2⟩ := eleven_dim_plane_of_no_sols_exists
+  let Y := Sum.elim B E
+  use Y
+  apply Fintype.linearIndependent_iff.mpr
+  intro g hg
+  rw [@Fintype.sum_sum_type] at hg
+  rw [@add_eq_zero_iff_eq_neg] at hg
+  rw [← @Finset.sum_neg_distrib] at hg
+  have h1 : ∑ x : Fin n, -(g (Sum.inr x) • Y (Sum.inr x)) =
+      ∑ x : Fin n, (-g (Sum.inr x)) • Y (Sum.inr x):= by
+    apply Finset.sum_congr
+    simp
+    intro i _
+    simp
+  rw [h1] at hg
+  have h2 : ∑ a₁ : Fin 11, g (Sum.inl a₁) • Y (Sum.inl a₁) = 0 := by
+    apply hB2
+    erw [hg]
+    exact hE2 fun i => -g (Sum.inr i)
+  rw [Fintype.linearIndependent_iff] at hB1 hE1
+  have h3 : ∀ i, g (Sum.inl i) = 0 := hB1 (fun i => (g (Sum.inl i))) h2
+  rw [h2] at hg
+  have h4 : ∀ i, - g (Sum.inr i) = 0 := hE1 (fun i => (- g (Sum.inr i))) hg.symm
+  simp at h4
+  intro i
+  match i with
+  | Sum.inl i => exact h3 i
+  | Sum.inr i => exact h4 i
+
+
+theorem plane_exists_dim_le_7 {n : ℕ} (hn : existsPlane n) : n ≤ 7 := by
+  obtain ⟨B, hB⟩ := exists_plane_exists_basis hn
+  have h1 := LinearIndependent.fintype_card_le_finrank hB
+  simp at h1
+  rw [show FiniteDimensional.finrank ℚ (PlusU1 3).charges = 18 from
+    FiniteDimensional.finrank_fin_fun ℚ] at h1
+  exact Nat.le_of_add_le_add_left h1
+
+
+end PlusU1
+end SMRHN