feat: Add basis in even and odd cases
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jstoobysmith
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@ -14,9 +14,11 @@ import HepLean.AnomalyCancellation.PureU1.Basic
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import HepLean.AnomalyCancellation.PureU1.BasisLinear
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import HepLean.AnomalyCancellation.PureU1.ConstAbs
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import HepLean.AnomalyCancellation.PureU1.LineInPlaneCond
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import HepLean.AnomalyCancellation.PureU1.Even.BasisLinear
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import HepLean.AnomalyCancellation.PureU1.LowDim.One
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import HepLean.AnomalyCancellation.PureU1.LowDim.Three
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import HepLean.AnomalyCancellation.PureU1.LowDim.Two
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import HepLean.AnomalyCancellation.PureU1.Odd.BasisLinear
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import HepLean.AnomalyCancellation.PureU1.Permutations
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import HepLean.AnomalyCancellation.PureU1.Sort
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import HepLean.AnomalyCancellation.PureU1.VectorLike
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756
HepLean/AnomalyCancellation/PureU1/Even/BasisLinear.lean
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756
HepLean/AnomalyCancellation/PureU1/Even/BasisLinear.lean
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@ -0,0 +1,756 @@
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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.PureU1.Sort
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import HepLean.AnomalyCancellation.PureU1.BasisLinear
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import HepLean.AnomalyCancellation.PureU1.VectorLike
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import Mathlib.Logic.Equiv.Fin
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/-!
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# Basis of `LinSols` in the even case
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We give a basis of `LinSols` in the even case. This basis has the special propoerty
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that splits into two planes on which every point is a solution to the ACCs.
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-/
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universe v u
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open Nat
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open Finset
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open BigOperators
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namespace PureU1
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variable {n : ℕ}
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namespace VectorLikeEvenPlane
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lemma n_cond₂ (n : ℕ) : 1 + ((n + n) + 1) = 2 * n.succ := by
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linarith
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section theδs
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/-- A helper function for what follows. -/
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def δ₁ (j : Fin n.succ) : Fin (2 * n.succ) := Fin.cast (split_equal n.succ) (Fin.castAdd n.succ j)
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/-- A helper function for what follows. -/
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def δ₂ (j : Fin n.succ) : Fin (2 * n.succ) := Fin.cast (split_equal n.succ) (Fin.natAdd n.succ j)
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/-- A helper function for what follows. -/
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def δ!₁ (j : Fin n) : Fin (2 * n.succ) := Fin.cast (n_cond₂ n)
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(Fin.natAdd 1 (Fin.castAdd 1 (Fin.castAdd n j)))
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/-- A helper function for what follows. -/
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def δ!₂ (j : Fin n) : Fin (2 * n.succ) := Fin.cast (n_cond₂ n)
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(Fin.natAdd 1 (Fin.castAdd 1 (Fin.natAdd n j)))
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/-- A helper function for what follows. -/
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def δ!₃ : Fin (2 * n.succ) := (Fin.cast (n_cond₂ n) (Fin.castAdd ((n + n) + 1) 0))
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/-- A helper function for what follows. -/
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def δ!₄ : Fin (2 * n.succ) := (Fin.cast (n_cond₂ n) (Fin.natAdd 1 (Fin.natAdd (n + n) 0)))
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lemma ext_δ (S T : Fin (2 * n.succ) → ℚ) (h1 : ∀ i, S (δ₁ i) = T (δ₁ i))
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(h2 : ∀ i, S (δ₂ i) = T (δ₂ i)) : S = T := by
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funext i
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by_cases hi : i.val < n.succ
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let j : Fin n.succ := ⟨i, hi⟩
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have h2 := h1 j
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have h3 : δ₁ j = i := by
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simp [δ₁, Fin.ext_iff]
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rw [h3] at h2
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exact h2
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let j : Fin n.succ := ⟨i - n.succ, by omega⟩
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have h2 := h2 j
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have h3 : δ₂ j = i := by
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simp [δ₂, Fin.ext_iff]
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omega
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rw [h3] at h2
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exact h2
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lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
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∑ i, S i = ∑ i : Fin n.succ, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
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have h1 : ∑ i, S i = ∑ i : Fin (n.succ + n.succ), S (Fin.cast (split_equal n.succ) i) := by
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rw [Finset.sum_equiv (Fin.castIso (split_equal n.succ)).symm.toEquiv]
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intro i
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simp
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intro i
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simp
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rw [h1]
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rw [Fin.sum_univ_add, Finset.sum_add_distrib]
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rfl
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lemma sum_δ₁_δ₂' (S : Fin (2 * n.succ) → ℚ) :
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∑ i, S i = ∑ i : Fin n.succ, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
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have h1 : ∑ i, S i = ∑ i : Fin (n.succ + n.succ), S (Fin.cast (split_equal n.succ) i) := by
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rw [Finset.sum_equiv (Fin.castIso (split_equal n.succ)).symm.toEquiv]
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intro i
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simp
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intro i
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simp
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rw [h1]
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rw [Fin.sum_univ_add, Finset.sum_add_distrib]
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rfl
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lemma sum_δ!₁_δ!₂ (S : Fin (2 * n.succ) → ℚ) :
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∑ i, S i = S δ!₃ + S δ!₄ + ∑ i : Fin n, ((S ∘ δ!₁) i + (S ∘ δ!₂) i) := by
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have h1 : ∑ i, S i = ∑ i : Fin (1 + ((n + n) + 1)), S (Fin.cast (n_cond₂ n) i) := by
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rw [Finset.sum_equiv (Fin.castIso (n_cond₂ n)).symm.toEquiv]
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intro i
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simp
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intro i
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simp
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rw [h1]
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rw [Fin.sum_univ_add, Fin.sum_univ_add, Fin.sum_univ_add, Finset.sum_add_distrib]
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simp
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repeat rw [Rat.add_assoc]
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apply congrArg
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rw [Rat.add_comm]
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rw [← Rat.add_assoc]
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nth_rewrite 2 [Rat.add_comm]
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repeat rw [Rat.add_assoc]
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nth_rewrite 2 [Rat.add_comm]
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rfl
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lemma δ!₃_δ₁0 : @δ!₃ n = δ₁ 0 := by
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rfl
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lemma δ!₄_δ₂Last: @δ!₄ n = δ₂ (Fin.last n) := by
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rw [Fin.ext_iff]
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simp [δ!₄, δ₂]
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omega
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lemma δ!₁_δ₁ (j : Fin n) : δ!₁ j = δ₁ j.succ := by
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rw [Fin.ext_iff, δ₁, δ!₁]
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simp
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ring
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lemma δ!₂_δ₂ (j : Fin n) : δ!₂ j = δ₂ j.castSucc := by
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rw [Fin.ext_iff, δ₂, δ!₂]
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simp
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ring_nf
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rw [Nat.succ_eq_add_one]
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ring
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end theδs
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/-- The first part of the basis as charges. -/
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def basisAsCharges (j : Fin n.succ) : (PureU1 (2 * n.succ)).charges :=
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fun i =>
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if i = δ₁ j then
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1
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else
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if i = δ₂ j then
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- 1
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else
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0
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/-- The second part of the basis as charges. -/
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def basis!AsCharges (j : Fin n) : (PureU1 (2 * n.succ)).charges :=
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fun i =>
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if i = δ!₁ j then
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1
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else
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if i = δ!₂ j then
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- 1
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else
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0
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lemma basis_on_δ₁_self (j : Fin n.succ) : basisAsCharges j (δ₁ j) = 1 := by
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simp [basisAsCharges]
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lemma basis!_on_δ!₁_self (j : Fin n) : basis!AsCharges j (δ!₁ j) = 1 := by
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simp [basis!AsCharges]
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lemma basis_on_δ₁_other {k j : Fin n.succ} (h : k ≠ j) :
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basisAsCharges k (δ₁ j) = 0 := by
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simp [basisAsCharges]
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simp [δ₁, δ₂]
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split
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rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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split
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rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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rfl
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lemma basis_on_other {k : Fin n.succ} {j : Fin (2 * n.succ)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
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basisAsCharges k j = 0 := by
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simp [basisAsCharges]
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simp_all only [ne_eq, ↓reduceIte]
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lemma basis!_on_other {k : Fin n} {j : Fin (2 * n.succ)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
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basis!AsCharges k j = 0 := by
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simp [basis!AsCharges]
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simp_all only [ne_eq, ↓reduceIte]
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lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
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basis!AsCharges k (δ!₁ j) = 0 := by
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simp [basis!AsCharges]
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simp [δ!₁, δ!₂]
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split
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rename_i h1
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rw [Fin.ext_iff] at h1
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simp_all
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rw [Fin.ext_iff] at h
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simp_all
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split
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rename_i h1 h2
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simp_all
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rw [Fin.ext_iff] at h2
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simp at h2
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omega
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rfl
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lemma basis_δ₂_eq_minus_δ₁ (j i : Fin n.succ) :
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basisAsCharges j (δ₂ i) = - basisAsCharges j (δ₁ i) := by
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simp [basisAsCharges, δ₂, δ₁]
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split <;> split
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any_goals split
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any_goals split
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any_goals rfl
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all_goals rename_i h1 h2
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all_goals rw [Fin.ext_iff] at h1 h2
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all_goals simp_all
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all_goals rename_i h3
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all_goals rw [Fin.ext_iff] at h3
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all_goals simp_all
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all_goals omega
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lemma basis!_δ!₂_eq_minus_δ!₁ (j i : Fin n) :
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basis!AsCharges j (δ!₂ i) = - basis!AsCharges j (δ!₁ i) := by
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simp [basis!AsCharges, δ!₂, δ!₁]
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split <;> split
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any_goals split
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any_goals split
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any_goals rfl
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all_goals rename_i h1 h2
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all_goals rw [Fin.ext_iff] at h1 h2
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all_goals simp_all
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subst h1
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exact Fin.elim0 i
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all_goals rename_i h3
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all_goals rw [Fin.ext_iff] at h3
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all_goals simp_all
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all_goals omega
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lemma basis_on_δ₂_self (j : Fin n.succ) : basisAsCharges j (δ₂ j) = - 1 := by
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rw [basis_δ₂_eq_minus_δ₁, basis_on_δ₁_self]
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lemma basis!_on_δ!₂_self (j : Fin n) : basis!AsCharges j (δ!₂ j) = - 1 := by
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rw [basis!_δ!₂_eq_minus_δ!₁, basis!_on_δ!₁_self]
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lemma basis_on_δ₂_other {k j : Fin n.succ} (h : k ≠ j) : basisAsCharges k (δ₂ j) = 0 := by
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rw [basis_δ₂_eq_minus_δ₁, basis_on_δ₁_other h]
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rfl
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lemma basis!_on_δ!₂_other {k j : Fin n} (h : k ≠ j) : basis!AsCharges k (δ!₂ j) = 0 := by
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rw [basis!_δ!₂_eq_minus_δ!₁, basis!_on_δ!₁_other h]
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rfl
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lemma basis!_on_δ!₃ (j : Fin n) : basis!AsCharges j δ!₃ = 0 := by
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simp [basis!AsCharges]
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split <;> rename_i h
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rw [Fin.ext_iff] at h
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simp [δ!₃, δ!₁] at h
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omega
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split <;> rename_i h2
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rw [Fin.ext_iff] at h2
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simp [δ!₃, δ!₂] at h2
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omega
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rfl
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lemma basis!_on_δ!₄ (j : Fin n) : basis!AsCharges j δ!₄ = 0 := by
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simp [basis!AsCharges]
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split <;> rename_i h
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rw [Fin.ext_iff] at h
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simp [δ!₄, δ!₁] at h
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omega
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split <;> rename_i h2
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rw [Fin.ext_iff] at h2
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simp [δ!₄, δ!₂] at h2
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omega
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rfl
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lemma basis_linearACC (j : Fin n.succ) : (accGrav (2 * n.succ)) (basisAsCharges j) = 0 := by
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rw [accGrav]
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simp
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rw [sum_δ₁_δ₂]
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simp [basis_δ₂_eq_minus_δ₁]
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lemma basis!_linearACC (j : Fin n) : (accGrav (2 * n.succ)) (basis!AsCharges j) = 0 := by
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rw [accGrav]
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simp
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rw [sum_δ!₁_δ!₂, basis!_on_δ!₃, basis!_on_δ!₄]
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simp [basis!_δ!₂_eq_minus_δ!₁]
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lemma basis_accCube (j : Fin n.succ) :
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accCube (2 * n.succ) (basisAsCharges j) = 0 := by
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rw [accCube_explicit, sum_δ₁_δ₂]
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apply Finset.sum_eq_zero
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intro i _
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simp [basis_δ₂_eq_minus_δ₁]
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ring
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lemma basis!_accCube (j : Fin n) :
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accCube (2 * n.succ) (basis!AsCharges j) = 0 := by
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rw [accCube_explicit, sum_δ!₁_δ!₂]
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rw [basis!_on_δ!₄, basis!_on_δ!₃]
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simp
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apply Finset.sum_eq_zero
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intro i _
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simp [basis!_δ!₂_eq_minus_δ!₁]
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ring
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/-- The first part of the basis as `LinSols`. -/
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@[simps!]
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def basis (j : Fin n.succ) : (PureU1 (2 * n.succ)).LinSols :=
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⟨basisAsCharges j, by
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intro i
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simp at i
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match i with
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| 0 =>
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exact basis_linearACC j⟩
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/-- The second part of the basis as `LinSols`. -/
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@[simps!]
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def basis! (j : Fin n) : (PureU1 (2 * n.succ)).LinSols :=
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⟨basis!AsCharges j, by
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intro i
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simp at i
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match i with
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| 0 =>
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exact basis!_linearACC j⟩
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/-- The whole basis as `LinSols`. -/
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def basisa : (Fin n.succ) ⊕ (Fin n) → (PureU1 (2 * n.succ)).LinSols := fun i =>
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match i with
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| .inl i => basis i
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| .inr i => basis! i
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/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/
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lemma swap!_as_add {S S' : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
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(hS : ((FamilyPermutations (2 * n.succ)).linSolRep
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(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
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S'.val = S.val + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j := by
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funext i
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rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
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by_cases hi : i = δ!₁ j
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subst hi
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simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
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by_cases hi2 : i = δ!₂ j
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subst hi2
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simp [HSMul.hSMul, basis!_on_δ!₂_self, pairSwap_inv_snd]
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simp [HSMul.hSMul]
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rw [basis!_on_other hi hi2]
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change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
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erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
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simp
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/-- A point in the span of the first part of the basis as a charge. -/
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def P (f : Fin n.succ → ℚ) : (PureU1 (2 * n.succ)).charges := ∑ i, f i • basisAsCharges i
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/-- A point in the span of the second part of the basis as a charge. -/
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def P! (f : Fin n → ℚ) : (PureU1 (2 * n.succ)).charges := ∑ i, f i • basis!AsCharges i
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/-- A point in the span of the basis as a charge. -/
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def Pa (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : (PureU1 (2 * n.succ)).charges := P f + P! g
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lemma P_δ₁ (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (δ₁ j) = f j := by
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rw [P, sum_of_charges]
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simp [HSMul.hSMul, SMul.smul]
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rw [Finset.sum_eq_single j]
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rw [basis_on_δ₁_self]
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simp
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intro k _ hkj
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rw [basis_on_δ₁_other hkj]
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simp only [mul_zero]
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simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
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|
||||
lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis!_on_δ!₁_self]
|
||||
simp
|
||||
intro k _ hkj
|
||||
rw [basis!_on_δ!₁_other hkj]
|
||||
simp only [mul_zero]
|
||||
simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
|
||||
|
||||
lemma Pa_δ!₁ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) (j : Fin n) : Pa f g (δ!₁ j) = f j.succ + g j := by
|
||||
rw [Pa]
|
||||
simp
|
||||
rw [P!_δ!₁, δ!₁_δ₁, P_δ₁]
|
||||
|
||||
lemma P_δ₂ (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (δ₂ j) = - f j := by
|
||||
rw [P, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis_on_δ₂_self]
|
||||
simp
|
||||
intro k _ hkj
|
||||
rw [basis_on_δ₂_other hkj]
|
||||
simp
|
||||
simp
|
||||
|
||||
lemma P!_δ!₂ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₂ j) = - f j := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis!_on_δ!₂_self]
|
||||
simp
|
||||
intro k _ hkj
|
||||
rw [basis!_on_δ!₂_other hkj]
|
||||
simp
|
||||
simp
|
||||
|
||||
lemma Pa_δ!₂ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) (j : Fin n) :
|
||||
Pa f g (δ!₂ j) = - f j.castSucc - g j := by
|
||||
rw [Pa]
|
||||
simp
|
||||
rw [P!_δ!₂, δ!₂_δ₂, P_δ₂]
|
||||
ring
|
||||
|
||||
lemma P!_δ!₃ (f : Fin n → ℚ) : P! f (δ!₃) = 0 := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul, basis!_on_δ!₃]
|
||||
|
||||
lemma Pa_δ!₃ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₃) = f 0 := by
|
||||
rw [Pa]
|
||||
simp
|
||||
rw [P!_δ!₃, δ!₃_δ₁0, P_δ₁]
|
||||
simp
|
||||
|
||||
lemma P!_δ!₄ (f : Fin n → ℚ) : P! f (δ!₄) = 0 := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul, basis!_on_δ!₄]
|
||||
|
||||
lemma Pa_δ!₄ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₄) = - f (Fin.last n) := by
|
||||
rw [Pa]
|
||||
simp
|
||||
rw [P!_δ!₄, δ!₄_δ₂Last, P_δ₂]
|
||||
simp
|
||||
|
||||
lemma P_δ₁_δ₂ (f : Fin n.succ → ℚ) : P f ∘ δ₂ = - P f ∘ δ₁ := by
|
||||
funext j
|
||||
simp
|
||||
rw [P_δ₁, P_δ₂]
|
||||
|
||||
lemma P_linearACC (f : Fin n.succ → ℚ) : (accGrav (2 * n.succ)) (P f) = 0 := by
|
||||
rw [accGrav]
|
||||
simp
|
||||
rw [sum_δ₁_δ₂]
|
||||
simp [P_δ₂, P_δ₁]
|
||||
|
||||
lemma P_accCube (f : Fin n.succ → ℚ) : accCube (2 * n.succ) (P f) = 0 := by
|
||||
rw [accCube_explicit, sum_δ₁_δ₂]
|
||||
apply Finset.sum_eq_zero
|
||||
intro i _
|
||||
simp [P_δ₁, P_δ₂]
|
||||
ring
|
||||
|
||||
lemma P!_accCube (f : Fin n → ℚ) : accCube (2 * n.succ) (P! f) = 0 := by
|
||||
rw [accCube_explicit, sum_δ!₁_δ!₂, P!_δ!₃, P!_δ!₄]
|
||||
simp
|
||||
apply Finset.sum_eq_zero
|
||||
intro i _
|
||||
simp [P!_δ!₁, P!_δ!₂]
|
||||
ring
|
||||
|
||||
lemma P_P_P!_accCube (g : Fin n.succ → ℚ) (j : Fin n) :
|
||||
accCubeTriLinSymm.toFun (P g, P g, basis!AsCharges j)
|
||||
= g (j.succ) ^ 2 - g (j.castSucc) ^ 2 := by
|
||||
simp [accCubeTriLinSymm]
|
||||
rw [sum_δ!₁_δ!₂, basis!_on_δ!₃, basis!_on_δ!₄]
|
||||
simp
|
||||
rw [Finset.sum_eq_single j, basis!_on_δ!₁_self, basis!_on_δ!₂_self]
|
||||
simp [δ!₁_δ₁, δ!₂_δ₂]
|
||||
rw [P_δ₁, P_δ₂]
|
||||
ring
|
||||
intro k _ hkj
|
||||
erw [basis!_on_δ!₁_other hkj.symm, basis!_on_δ!₂_other hkj.symm]
|
||||
simp
|
||||
simp
|
||||
|
||||
lemma P_P!_P!_accCube (g : Fin n → ℚ) (j : Fin n.succ) :
|
||||
accCubeTriLinSymm.toFun (P! g, P! g, basisAsCharges j)
|
||||
= (P! g (δ₁ j))^2 - (P! g (δ₂ j))^2 := by
|
||||
simp [accCubeTriLinSymm]
|
||||
rw [sum_δ₁_δ₂]
|
||||
simp
|
||||
rw [Finset.sum_eq_single j, basis_on_δ₁_self, basis_on_δ₂_self]
|
||||
simp [δ!₁_δ₁, δ!₂_δ₂]
|
||||
ring
|
||||
intro k _ hkj
|
||||
erw [basis_on_δ₁_other hkj.symm, basis_on_δ₂_other hkj.symm]
|
||||
simp
|
||||
simp
|
||||
|
||||
lemma P_zero (f : Fin n.succ → ℚ) (h : P f = 0) : ∀ i, f i = 0 := by
|
||||
intro i
|
||||
erw [← P_δ₁ f]
|
||||
rw [h]
|
||||
rfl
|
||||
|
||||
lemma P!_zero (f : Fin n → ℚ) (h : P! f = 0) : ∀ i, f i = 0 := by
|
||||
intro i
|
||||
rw [← P!_δ!₁ f]
|
||||
rw [h]
|
||||
rfl
|
||||
|
||||
lemma Pa_zero (f : Fin n.succ → ℚ) (g : Fin n → ℚ) (h : Pa f g = 0) :
|
||||
∀ i, f i = 0 := by
|
||||
have h₃ := Pa_δ!₃ f g
|
||||
rw [h] at h₃
|
||||
change 0 = f 0 at h₃
|
||||
intro i
|
||||
have hinduc (iv : ℕ) (hiv : iv < n.succ) : f ⟨iv, hiv⟩ = 0 := by
|
||||
induction iv
|
||||
exact h₃.symm
|
||||
rename_i iv hi
|
||||
have hivi : iv < n.succ := by omega
|
||||
have hi2 := hi hivi
|
||||
have h1 := Pa_δ!₁ f g ⟨iv, by omega⟩
|
||||
have h2 := Pa_δ!₂ f g ⟨iv, by omega⟩
|
||||
rw [h] at h1 h2
|
||||
simp at h1 h2
|
||||
erw [hi2] at h2
|
||||
change 0 = _ at h2
|
||||
simp at h2
|
||||
rw [h2] at h1
|
||||
simp at h1
|
||||
exact h1.symm
|
||||
exact hinduc i.val i.prop
|
||||
|
||||
lemma Pa_zero! (f : Fin n.succ → ℚ) (g : Fin n → ℚ) (h : Pa f g = 0) :
|
||||
∀ i, g i = 0 := by
|
||||
have hf := Pa_zero f g h
|
||||
rw [Pa, P] at h
|
||||
simp [hf] at h
|
||||
exact P!_zero g h
|
||||
|
||||
/-- A point in the span of the first part of the basis. -/
|
||||
def P' (f : Fin n.succ → ℚ) : (PureU1 (2 * n.succ)).LinSols := ∑ i, f i • basis i
|
||||
|
||||
/-- A point in the span of the second part of the basis. -/
|
||||
def P!' (f : Fin n → ℚ) : (PureU1 (2 * n.succ)).LinSols := ∑ i, f i • basis! i
|
||||
|
||||
/-- A point in the span of the whole basis. -/
|
||||
def Pa' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n.succ)).LinSols :=
|
||||
∑ i, f i • basisa i
|
||||
|
||||
lemma Pa'_P'_P!' (f : (Fin n.succ) ⊕ (Fin n) → ℚ) :
|
||||
Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr):= by
|
||||
exact Fintype.sum_sum_type _
|
||||
|
||||
lemma P'_val (f : Fin n.succ → ℚ) : (P' f).val = P f := by
|
||||
simp [P', P]
|
||||
funext i
|
||||
rw [sum_of_anomaly_free_linear, sum_of_charges]
|
||||
simp [HSMul.hSMul]
|
||||
|
||||
lemma P!'_val (f : Fin n → ℚ) : (P!' f).val = P! f := by
|
||||
simp [P!', P!]
|
||||
funext i
|
||||
rw [sum_of_anomaly_free_linear, sum_of_charges]
|
||||
simp [HSMul.hSMul]
|
||||
|
||||
theorem basis_linear_independent : LinearIndependent ℚ (@basis n) := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
change P' f = 0 at h
|
||||
have h1 : (P' f).val = 0 := by
|
||||
simp [h]
|
||||
rfl
|
||||
rw [P'_val] at h1
|
||||
exact P_zero f h1
|
||||
|
||||
theorem basis!_linear_independent : LinearIndependent ℚ (@basis! n) := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
change P!' f = 0 at h
|
||||
have h1 : (P!' f).val = 0 := by
|
||||
simp [h]
|
||||
rfl
|
||||
rw [P!'_val] at h1
|
||||
exact P!_zero f h1
|
||||
|
||||
|
||||
theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n) := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
change Pa' f = 0 at h
|
||||
have h1 : (Pa' f).val = 0 := by
|
||||
simp [h]
|
||||
rfl
|
||||
rw [Pa'_P'_P!'] at h1
|
||||
change (P' (f ∘ Sum.inl)).val + (P!' (f ∘ Sum.inr)).val = 0 at h1
|
||||
rw [P!'_val, P'_val] at h1
|
||||
change Pa (f ∘ Sum.inl) (f ∘ Sum.inr) = 0 at h1
|
||||
have hf := Pa_zero (f ∘ Sum.inl) (f ∘ Sum.inr) h1
|
||||
have hg := Pa_zero! (f ∘ Sum.inl) (f ∘ Sum.inr) h1
|
||||
intro i
|
||||
simp_all
|
||||
cases i
|
||||
simp_all
|
||||
simp_all
|
||||
|
||||
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
funext i
|
||||
rw [Pa', Pa'] at h
|
||||
have h1 : ∑ i : Fin (succ n) ⊕ Fin n, (f i + (- f' i)) • basisa i = 0 := by
|
||||
simp only [add_smul, neg_smul]
|
||||
rw [Finset.sum_add_distrib]
|
||||
rw [h]
|
||||
rw [← Finset.sum_add_distrib]
|
||||
simp
|
||||
have h2 : ∀ i, (f i + (- f' i)) = 0 := by
|
||||
exact Fintype.linearIndependent_iff.mp (@basisa_linear_independent (n))
|
||||
(fun i => f i + -f' i) h1
|
||||
have h2i := h2 i
|
||||
linarith
|
||||
intro h
|
||||
rw [h]
|
||||
|
||||
/-- A helper function for what follows. TODO: replace this with mathlib functions. -/
|
||||
def join (g : Fin n.succ → ℚ) (f : Fin n → ℚ) : (Fin n.succ) ⊕ (Fin n) → ℚ := fun i =>
|
||||
match i with
|
||||
| .inl i => g i
|
||||
| .inr i => f i
|
||||
|
||||
lemma join_ext (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
|
||||
join g f = join g' f' ↔ g = g' ∧ f = f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
apply And.intro
|
||||
funext i
|
||||
exact congr_fun h (Sum.inl i)
|
||||
funext i
|
||||
exact congr_fun h (Sum.inr i)
|
||||
intro h
|
||||
rw [h.left, h.right]
|
||||
|
||||
lemma join_Pa (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
|
||||
Pa' (join g f) = Pa' (join g' f') ↔ Pa g f = Pa g' f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [Pa'_eq] at h
|
||||
rw [join_ext] at h
|
||||
rw [h.left, h.right]
|
||||
intro h
|
||||
apply ACCSystemLinear.LinSols.ext
|
||||
rw [Pa'_P'_P!', Pa'_P'_P!']
|
||||
simp [P'_val, P!'_val]
|
||||
exact h
|
||||
|
||||
lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
|
||||
Pa g f = Pa g' f' ↔ g = g' ∧ f = f' := by
|
||||
rw [← join_Pa]
|
||||
rw [← join_ext]
|
||||
exact Pa'_eq _ _
|
||||
|
||||
|
||||
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n)) =
|
||||
FiniteDimensional.finrank ℚ (PureU1 (2 * n.succ)).LinSols := by
|
||||
erw [BasisLinear.finrank_AnomalyFreeLinear]
|
||||
simp
|
||||
omega
|
||||
|
||||
/-- The basis formed out of our basisa vectors. -/
|
||||
noncomputable def basisaAsBasis :
|
||||
Basis (Fin (succ n) ⊕ Fin n) ℚ (PureU1 (2 * succ n)).LinSols :=
|
||||
basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card
|
||||
|
||||
|
||||
lemma span_basis (S : (PureU1 (2 * n.succ)).LinSols) :
|
||||
∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f := by
|
||||
have h := (mem_span_range_iff_exists_fun ℚ).mp (Basis.mem_span basisaAsBasis S)
|
||||
obtain ⟨f, hf⟩ := h
|
||||
simp [basisaAsBasis] at hf
|
||||
change P' _ + P!' _ = S at hf
|
||||
use f ∘ Sum.inl
|
||||
use f ∘ Sum.inr
|
||||
rw [← hf]
|
||||
simp [P'_val, P!'_val]
|
||||
rfl
|
||||
|
||||
lemma P!_in_span (f : Fin n → ℚ) : P! f ∈ Submodule.span ℚ (Set.range basis!AsCharges) := by
|
||||
rw [(mem_span_range_iff_exists_fun ℚ)]
|
||||
use f
|
||||
rfl
|
||||
|
||||
lemma smul_basis!AsCharges_in_span (S : (PureU1 (2 * n.succ )).LinSols) (j : Fin n) :
|
||||
(S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j ∈
|
||||
Submodule.span ℚ (Set.range basis!AsCharges) := by
|
||||
apply Submodule.smul_mem
|
||||
apply SetLike.mem_of_subset
|
||||
apply Submodule.subset_span
|
||||
simp_all only [Set.mem_range, exists_apply_eq_apply]
|
||||
|
||||
lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
|
||||
(hS : ((FamilyPermutations (2 * n.succ)).linSolRep
|
||||
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') (g : Fin n.succ → ℚ) (f : Fin n → ℚ)
|
||||
(h : S.val = P g + P! f):
|
||||
∃
|
||||
(g' : Fin n.succ → ℚ) (f' : Fin n → ℚ) ,
|
||||
S'.val = P g' + P! f' ∧ P! f' = P! f +
|
||||
(S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j ∧ g' = g := by
|
||||
let X := P! f + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j
|
||||
have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
|
||||
apply Submodule.add_mem
|
||||
exact (P!_in_span f)
|
||||
exact (smul_basis!AsCharges_in_span S j)
|
||||
have hXsum := (mem_span_range_iff_exists_fun ℚ).mp hX
|
||||
obtain ⟨f', hf'⟩ := hXsum
|
||||
use g
|
||||
use f'
|
||||
change P! f' = _ at hf'
|
||||
erw [hf']
|
||||
simp
|
||||
change S'.val = P g + (P! f + _)
|
||||
rw [← add_assoc, ← h]
|
||||
apply swap!_as_add at hS
|
||||
exact hS
|
||||
|
||||
lemma vectorLikeEven_in_span (S : (PureU1 (2 * n.succ)).LinSols)
|
||||
(hS : vectorLikeEven S.val) :
|
||||
∃ (M : (FamilyPermutations (2 * n.succ)).group),
|
||||
(FamilyPermutations (2 * n.succ)).linSolRep M S
|
||||
∈ Submodule.span ℚ (Set.range basis) := by
|
||||
use (Tuple.sort S.val).symm
|
||||
change sortAFL S ∈ Submodule.span ℚ (Set.range basis)
|
||||
rw [mem_span_range_iff_exists_fun ℚ]
|
||||
let f : Fin n.succ → ℚ := fun i => (sortAFL S).val (δ₁ i)
|
||||
use f
|
||||
apply ACCSystemLinear.LinSols.ext
|
||||
rw [sortAFL_val]
|
||||
erw [P'_val]
|
||||
apply ext_δ
|
||||
intro i
|
||||
rw [P_δ₁]
|
||||
rfl
|
||||
intro i
|
||||
rw [P_δ₂]
|
||||
have ht := hS i
|
||||
change sort S.val (δ₁ i) = - sort S.val (δ₂ i) at ht
|
||||
have h : sort S.val (δ₂ i) = - sort S.val (δ₁ i) := by
|
||||
rw [ht]
|
||||
ring
|
||||
rw [h]
|
||||
simp
|
||||
rfl
|
||||
|
||||
end VectorLikeEvenPlane
|
||||
|
||||
end PureU1
|
||||
713
HepLean/AnomalyCancellation/PureU1/Odd/BasisLinear.lean
Normal file
713
HepLean/AnomalyCancellation/PureU1/Odd/BasisLinear.lean
Normal file
|
|
@ -0,0 +1,713 @@
|
|||
/-
|
||||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||||
Released under Apache 2.0 license.
|
||||
Authors: Joseph Tooby-Smith
|
||||
-/
|
||||
import HepLean.AnomalyCancellation.PureU1.Sort
|
||||
import HepLean.AnomalyCancellation.PureU1.BasisLinear
|
||||
import HepLean.AnomalyCancellation.PureU1.VectorLike
|
||||
import Mathlib.Logic.Equiv.Fin
|
||||
/-!
|
||||
# Basis of `LinSols` in the odd case
|
||||
|
||||
We give a basis of `LinSols` in the odd case. This basis has the special propoerty
|
||||
that splits into two planes on which every point is a solution to the ACCs.
|
||||
-/
|
||||
universe v u
|
||||
|
||||
open Nat
|
||||
open Finset
|
||||
open BigOperators
|
||||
|
||||
namespace PureU1
|
||||
|
||||
variable {n : ℕ}
|
||||
|
||||
|
||||
namespace VectorLikeOddPlane
|
||||
|
||||
lemma split_odd! (n : ℕ) : (1 + n) + n = 2 * n +1 := by
|
||||
omega
|
||||
|
||||
lemma split_adda (n : ℕ) : ((1+n)+1) + n.succ = 2 * n.succ + 1 := by
|
||||
omega
|
||||
|
||||
section theDeltas
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δ₁ (j : Fin n) : Fin (2 * n + 1) :=
|
||||
Fin.cast (split_odd n) (Fin.castAdd n (Fin.castAdd 1 j))
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δ₂ (j : Fin n) : Fin (2 * n + 1) :=
|
||||
Fin.cast (split_odd n) (Fin.natAdd (n+1) j)
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δ₃ : Fin (2 * n + 1) :=
|
||||
Fin.cast (split_odd n) (Fin.castAdd n (Fin.natAdd n 1))
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δ!₁ (j : Fin n) : Fin (2 * n + 1) :=
|
||||
Fin.cast (split_odd! n) (Fin.castAdd n (Fin.natAdd 1 j))
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δ!₂ (j : Fin n) : Fin (2 * n + 1) :=
|
||||
Fin.cast (split_odd! n) (Fin.natAdd (1 + n) j)
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δ!₃ : Fin (2 * n + 1) :=
|
||||
Fin.cast (split_odd! n) (Fin.castAdd n (Fin.castAdd n 1))
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δa₁ : Fin (2 * n.succ + 1) :=
|
||||
Fin.cast (split_adda n) (Fin.castAdd n.succ (Fin.castAdd 1 (Fin.castAdd n 1)))
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δa₂ (j : Fin n) : Fin (2 * n.succ + 1) :=
|
||||
Fin.cast (split_adda n) (Fin.castAdd n.succ (Fin.castAdd 1 (Fin.natAdd 1 j)))
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δa₃ : Fin (2 * n.succ + 1) :=
|
||||
Fin.cast (split_adda n) (Fin.castAdd n.succ (Fin.natAdd (1+n) 1))
|
||||
|
||||
/-- A helper function for what follows. -/
|
||||
def δa₄ (j : Fin n.succ) : Fin (2 * n.succ + 1) :=
|
||||
Fin.cast (split_adda n) (Fin.natAdd ((1+n)+1) j)
|
||||
|
||||
lemma δa₁_δ₁ : @δa₁ n = δ₁ 0 := by
|
||||
rfl
|
||||
|
||||
lemma δa₁_δ!₃ : @δa₁ n = δ!₃ := by
|
||||
rfl
|
||||
|
||||
lemma δa₂_δ₁ (j : Fin n) : δa₂ j = δ₁ j.succ := by
|
||||
rw [Fin.ext_iff]
|
||||
simp [δa₂, δ₁]
|
||||
omega
|
||||
|
||||
lemma δa₂_δ!₁ (j : Fin n) : δa₂ j = δ!₁ j.castSucc := by
|
||||
rw [Fin.ext_iff]
|
||||
simp [δa₂, δ!₁]
|
||||
|
||||
lemma δa₃_δ₃ : @δa₃ n = δ₃ := by
|
||||
rw [Fin.ext_iff]
|
||||
simp [δa₃, δ₃]
|
||||
omega
|
||||
|
||||
lemma δa₃_δ!₁ : δa₃ = δ!₁ (Fin.last n) := by
|
||||
rfl
|
||||
|
||||
lemma δa₄_δ₂ (j : Fin n.succ) : δa₄ j = δ₂ j := by
|
||||
rw [Fin.ext_iff]
|
||||
simp [δa₄, δ₂]
|
||||
omega
|
||||
|
||||
lemma δa₄_δ!₂ (j : Fin n.succ) : δa₄ j = δ!₂ j := by
|
||||
rw [Fin.ext_iff]
|
||||
simp [δa₄, δ!₂]
|
||||
omega
|
||||
|
||||
lemma δ₂_δ!₂ (j : Fin n) : δ₂ j = δ!₂ j := by
|
||||
rw [Fin.ext_iff]
|
||||
simp [δ₂, δ!₂]
|
||||
omega
|
||||
|
||||
lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
|
||||
∑ i, S i = S δ₃ + ∑ i : Fin n, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
|
||||
have h1 : ∑ i, S i = ∑ i : Fin (n + 1 + n), S (Fin.cast (split_odd n) i) := by
|
||||
rw [Finset.sum_equiv (Fin.castIso (split_odd n)).symm.toEquiv]
|
||||
intro i
|
||||
simp
|
||||
intro i
|
||||
simp
|
||||
rw [h1]
|
||||
rw [Fin.sum_univ_add, Fin.sum_univ_add]
|
||||
simp
|
||||
nth_rewrite 2 [add_comm]
|
||||
rw [add_assoc]
|
||||
rw [Finset.sum_add_distrib]
|
||||
rfl
|
||||
|
||||
lemma sum_δ! (S : Fin (2 * n + 1) → ℚ) :
|
||||
∑ i, S i = S δ!₃ + ∑ i : Fin n, ((S ∘ δ!₁) i + (S ∘ δ!₂) i) := by
|
||||
have h1 : ∑ i, S i = ∑ i : Fin ((1+n)+n), S (Fin.cast (split_odd! n) i) := by
|
||||
rw [Finset.sum_equiv (Fin.castIso (split_odd! n)).symm.toEquiv]
|
||||
intro i
|
||||
simp
|
||||
intro i
|
||||
simp
|
||||
rw [h1]
|
||||
rw [Fin.sum_univ_add, Fin.sum_univ_add]
|
||||
simp
|
||||
rw [add_assoc]
|
||||
rw [Finset.sum_add_distrib]
|
||||
rfl
|
||||
|
||||
end theDeltas
|
||||
|
||||
section theBasisVectors
|
||||
|
||||
/-- The first part of the basis as charge assignments. -/
|
||||
def basisAsCharges (j : Fin n) : (PureU1 (2 * n + 1)).charges :=
|
||||
fun i =>
|
||||
if i = δ₁ j then
|
||||
1
|
||||
else
|
||||
if i = δ₂ j then
|
||||
- 1
|
||||
else
|
||||
0
|
||||
|
||||
/-- The second part of the basis as charge assignments. -/
|
||||
def basis!AsCharges (j : Fin n) : (PureU1 (2 * n + 1)).charges :=
|
||||
fun i =>
|
||||
if i = δ!₁ j then
|
||||
1
|
||||
else
|
||||
if i = δ!₂ j then
|
||||
- 1
|
||||
else
|
||||
0
|
||||
|
||||
lemma basis_on_δ₁_self (j : Fin n) : basisAsCharges j (δ₁ j) = 1 := by
|
||||
simp [basisAsCharges]
|
||||
|
||||
lemma basis!_on_δ!₁_self (j : Fin n) : basis!AsCharges j (δ!₁ j) = 1 := by
|
||||
simp [basis!AsCharges]
|
||||
|
||||
lemma basis_on_δ₁_other {k j : Fin n} (h : k ≠ j) :
|
||||
basisAsCharges k (δ₁ j) = 0 := by
|
||||
simp [basisAsCharges]
|
||||
simp [δ₁, δ₂]
|
||||
split
|
||||
rename_i h1
|
||||
rw [Fin.ext_iff] at h1
|
||||
simp_all
|
||||
rw [Fin.ext_iff] at h
|
||||
simp_all
|
||||
split
|
||||
rename_i h1 h2
|
||||
simp_all
|
||||
rw [Fin.ext_iff] at h2
|
||||
simp at h2
|
||||
omega
|
||||
rfl
|
||||
|
||||
lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
|
||||
basis!AsCharges k (δ!₁ j) = 0 := by
|
||||
simp [basis!AsCharges]
|
||||
simp [δ!₁, δ!₂]
|
||||
split
|
||||
rename_i h1
|
||||
rw [Fin.ext_iff] at h1
|
||||
simp_all
|
||||
rw [Fin.ext_iff] at h
|
||||
simp_all
|
||||
split
|
||||
rename_i h1 h2
|
||||
simp_all
|
||||
rw [Fin.ext_iff] at h2
|
||||
simp at h2
|
||||
omega
|
||||
rfl
|
||||
|
||||
lemma basis_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
|
||||
basisAsCharges k j = 0 := by
|
||||
simp [basisAsCharges]
|
||||
simp_all only [ne_eq, ↓reduceIte]
|
||||
|
||||
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
|
||||
basis!AsCharges k j = 0 := by
|
||||
simp [basis!AsCharges]
|
||||
simp_all only [ne_eq, ↓reduceIte]
|
||||
|
||||
lemma basis_δ₂_eq_minus_δ₁ (j i : Fin n) :
|
||||
basisAsCharges j (δ₂ i) = - basisAsCharges j (δ₁ i) := by
|
||||
simp [basisAsCharges, δ₂, δ₁]
|
||||
split <;> split
|
||||
any_goals split
|
||||
any_goals split
|
||||
any_goals rfl
|
||||
all_goals rename_i h1 h2
|
||||
all_goals rw [Fin.ext_iff] at h1 h2
|
||||
all_goals simp_all
|
||||
all_goals rename_i h3
|
||||
all_goals rw [Fin.ext_iff] at h3
|
||||
all_goals simp_all
|
||||
all_goals omega
|
||||
|
||||
lemma basis!_δ!₂_eq_minus_δ!₁ (j i : Fin n) :
|
||||
basis!AsCharges j (δ!₂ i) = - basis!AsCharges j (δ!₁ i) := by
|
||||
simp [basis!AsCharges, δ!₂, δ!₁]
|
||||
split <;> split
|
||||
any_goals split
|
||||
any_goals split
|
||||
any_goals rfl
|
||||
all_goals rename_i h1 h2
|
||||
all_goals rw [Fin.ext_iff] at h1 h2
|
||||
all_goals simp_all
|
||||
subst h1
|
||||
exact Fin.elim0 i
|
||||
all_goals rename_i h3
|
||||
all_goals rw [Fin.ext_iff] at h3
|
||||
all_goals simp_all
|
||||
all_goals omega
|
||||
|
||||
lemma basis_on_δ₂_self (j : Fin n) : basisAsCharges j (δ₂ j) = - 1 := by
|
||||
rw [basis_δ₂_eq_minus_δ₁, basis_on_δ₁_self]
|
||||
|
||||
lemma basis!_on_δ!₂_self (j : Fin n) : basis!AsCharges j (δ!₂ j) = - 1 := by
|
||||
rw [basis!_δ!₂_eq_minus_δ!₁, basis!_on_δ!₁_self]
|
||||
|
||||
lemma basis_on_δ₂_other {k j : Fin n} (h : k ≠ j) : basisAsCharges k (δ₂ j) = 0 := by
|
||||
rw [basis_δ₂_eq_minus_δ₁, basis_on_δ₁_other h]
|
||||
rfl
|
||||
|
||||
lemma basis!_on_δ!₂_other {k j : Fin n} (h : k ≠ j) : basis!AsCharges k (δ!₂ j) = 0 := by
|
||||
rw [basis!_δ!₂_eq_minus_δ!₁, basis!_on_δ!₁_other h]
|
||||
rfl
|
||||
|
||||
lemma basis_on_δ₃ (j : Fin n) : basisAsCharges j δ₃ = 0 := by
|
||||
simp [basisAsCharges]
|
||||
split <;> rename_i h
|
||||
rw [Fin.ext_iff] at h
|
||||
simp [δ₃, δ₁] at h
|
||||
omega
|
||||
split <;> rename_i h2
|
||||
rw [Fin.ext_iff] at h2
|
||||
simp [δ₃, δ₂] at h2
|
||||
omega
|
||||
rfl
|
||||
|
||||
lemma basis!_on_δ!₃ (j : Fin n) : basis!AsCharges j δ!₃ = 0 := by
|
||||
simp [basis!AsCharges]
|
||||
split <;> rename_i h
|
||||
rw [Fin.ext_iff] at h
|
||||
simp [δ!₃, δ!₁] at h
|
||||
omega
|
||||
split <;> rename_i h2
|
||||
rw [Fin.ext_iff] at h2
|
||||
simp [δ!₃, δ!₂] at h2
|
||||
omega
|
||||
rfl
|
||||
|
||||
lemma basis_linearACC (j : Fin n) : (accGrav (2 * n + 1)) (basisAsCharges j) = 0 := by
|
||||
rw [accGrav]
|
||||
simp
|
||||
erw [sum_δ]
|
||||
simp [basis_δ₂_eq_minus_δ₁, basis_on_δ₃]
|
||||
|
||||
lemma basis!_linearACC (j : Fin n) : (accGrav (2 * n + 1)) (basis!AsCharges j) = 0 := by
|
||||
rw [accGrav]
|
||||
simp
|
||||
rw [sum_δ!, basis!_on_δ!₃]
|
||||
simp [basis!_δ!₂_eq_minus_δ!₁]
|
||||
|
||||
/-- The first part of the basis as `LinSols`. -/
|
||||
@[simps!]
|
||||
def basis (j : Fin n) : (PureU1 (2 * n + 1)).LinSols :=
|
||||
⟨basisAsCharges j, by
|
||||
intro i
|
||||
simp at i
|
||||
match i with
|
||||
| 0 =>
|
||||
exact basis_linearACC j⟩
|
||||
|
||||
/-- The second part of the basis as `LinSols`. -/
|
||||
@[simps!]
|
||||
def basis! (j : Fin n) : (PureU1 (2 * n + 1)).LinSols :=
|
||||
⟨basis!AsCharges j, by
|
||||
intro i
|
||||
simp at i
|
||||
match i with
|
||||
| 0 =>
|
||||
exact basis!_linearACC j⟩
|
||||
|
||||
/-- The whole basis as `LinSols`. -/
|
||||
def basisa : Fin n ⊕ Fin n → (PureU1 (2 * n + 1)).LinSols := fun i =>
|
||||
match i with
|
||||
| .inl i => basis i
|
||||
| .inr i => basis! i
|
||||
|
||||
end theBasisVectors
|
||||
|
||||
/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/
|
||||
lemma swap!_as_add {S S' : (PureU1 (2 * n + 1)).LinSols} (j : Fin n)
|
||||
(hS : ((FamilyPermutations (2 * n + 1)).linSolRep
|
||||
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
|
||||
S'.val = S.val + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j := by
|
||||
funext i
|
||||
rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
|
||||
by_cases hi : i = δ!₁ j
|
||||
subst hi
|
||||
simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
|
||||
by_cases hi2 : i = δ!₂ j
|
||||
subst hi2
|
||||
simp [HSMul.hSMul,basis!_on_δ!₂_self, pairSwap_inv_snd]
|
||||
simp [HSMul.hSMul]
|
||||
rw [basis!_on_other hi hi2]
|
||||
change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
|
||||
erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
|
||||
simp
|
||||
|
||||
/-- A point in the span of the first part of the basis as a charge. -/
|
||||
def P (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).charges := ∑ i, f i • basisAsCharges i
|
||||
|
||||
/-- A point in the span of the second part of the basis as a charge. -/
|
||||
def P! (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).charges := ∑ i, f i • basis!AsCharges i
|
||||
|
||||
/-- A point in the span of the basis as a charge. -/
|
||||
def Pa (f : Fin n → ℚ) (g : Fin n → ℚ) : (PureU1 (2 * n + 1)).charges := P f + P! g
|
||||
|
||||
lemma P_δ₁ (f : Fin n → ℚ) (j : Fin n) : P f (δ₁ j) = f j := by
|
||||
rw [P, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis_on_δ₁_self]
|
||||
simp
|
||||
intro k _ hkj
|
||||
rw [basis_on_δ₁_other hkj]
|
||||
simp only [mul_zero]
|
||||
simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
|
||||
|
||||
lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis!_on_δ!₁_self]
|
||||
simp
|
||||
intro k _ hkj
|
||||
rw [basis!_on_δ!₁_other hkj]
|
||||
simp only [mul_zero]
|
||||
simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
|
||||
|
||||
lemma P_δ₂ (f : Fin n → ℚ) (j : Fin n) : P f (δ₂ j) = - f j := by
|
||||
rw [P, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis_on_δ₂_self]
|
||||
simp
|
||||
intro k _ hkj
|
||||
rw [basis_on_δ₂_other hkj]
|
||||
simp
|
||||
simp
|
||||
|
||||
lemma P!_δ!₂ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₂ j) = - f j := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
rw [Finset.sum_eq_single j]
|
||||
rw [basis!_on_δ!₂_self]
|
||||
simp
|
||||
intro k _ hkj
|
||||
rw [basis!_on_δ!₂_other hkj]
|
||||
simp
|
||||
simp
|
||||
|
||||
lemma P_δ₃ (f : Fin n → ℚ) : P f (δ₃) = 0 := by
|
||||
rw [P, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul, basis_on_δ₃]
|
||||
|
||||
lemma P!_δ!₃ (f : Fin n → ℚ) : P! f (δ!₃) = 0 := by
|
||||
rw [P!, sum_of_charges]
|
||||
simp [HSMul.hSMul, SMul.smul, basis!_on_δ!₃]
|
||||
|
||||
lemma Pa_δa₁ (f g : Fin n.succ → ℚ) : Pa f g δa₁ = f 0 := by
|
||||
rw [Pa]
|
||||
simp
|
||||
nth_rewrite 1 [δa₁_δ₁]
|
||||
rw [δa₁_δ!₃]
|
||||
rw [P_δ₁, P!_δ!₃]
|
||||
simp
|
||||
|
||||
lemma Pa_δa₂ (f g : Fin n.succ → ℚ) (j : Fin n) : Pa f g (δa₂ j) = f j.succ + g j.castSucc := by
|
||||
rw [Pa]
|
||||
simp
|
||||
nth_rewrite 1 [δa₂_δ₁]
|
||||
rw [δa₂_δ!₁]
|
||||
rw [P_δ₁, P!_δ!₁]
|
||||
|
||||
lemma Pa_δa₃ (f g : Fin n.succ → ℚ) : Pa f g (δa₃) = g (Fin.last n) := by
|
||||
rw [Pa]
|
||||
simp
|
||||
nth_rewrite 1 [δa₃_δ₃]
|
||||
rw [δa₃_δ!₁]
|
||||
rw [P_δ₃, P!_δ!₁]
|
||||
simp
|
||||
|
||||
lemma Pa_δa₄ (f g : Fin n.succ → ℚ) (j : Fin n.succ) : Pa f g (δa₄ j) = - f j - g j := by
|
||||
rw [Pa]
|
||||
simp
|
||||
nth_rewrite 1 [δa₄_δ₂]
|
||||
rw [δa₄_δ!₂]
|
||||
rw [P_δ₂, P!_δ!₂]
|
||||
ring
|
||||
|
||||
lemma P_linearACC (f : Fin n → ℚ) : (accGrav (2 * n + 1)) (P f) = 0 := by
|
||||
rw [accGrav]
|
||||
simp
|
||||
rw [sum_δ]
|
||||
simp [P_δ₂, P_δ₁, P_δ₃]
|
||||
|
||||
lemma P!_linearACC (f : Fin n → ℚ) : (accGrav (2 * n + 1)) (P! f) = 0 := by
|
||||
rw [accGrav]
|
||||
simp
|
||||
rw [sum_δ!]
|
||||
simp [P!_δ!₂, P!_δ!₁, P!_δ!₃]
|
||||
|
||||
lemma P_accCube (f : Fin n → ℚ) : accCube (2 * n +1) (P f) = 0 := by
|
||||
rw [accCube_explicit, sum_δ, P_δ₃]
|
||||
simp
|
||||
apply Finset.sum_eq_zero
|
||||
intro i _
|
||||
simp [P_δ₁, P_δ₂]
|
||||
ring
|
||||
|
||||
lemma P!_accCube (f : Fin n → ℚ) : accCube (2 * n +1) (P! f) = 0 := by
|
||||
rw [accCube_explicit, sum_δ!, P!_δ!₃]
|
||||
simp
|
||||
apply Finset.sum_eq_zero
|
||||
intro i _
|
||||
simp [P!_δ!₁, P!_δ!₂]
|
||||
ring
|
||||
|
||||
lemma P_P_P!_accCube (g : Fin n → ℚ) (j : Fin n) :
|
||||
accCubeTriLinSymm.toFun (P g, P g, basis!AsCharges j)
|
||||
= (P g (δ!₁ j))^2 - (g j)^2 := by
|
||||
simp [accCubeTriLinSymm]
|
||||
rw [sum_δ!, basis!_on_δ!₃]
|
||||
simp
|
||||
rw [Finset.sum_eq_single j, basis!_on_δ!₁_self, basis!_on_δ!₂_self]
|
||||
rw [← δ₂_δ!₂, P_δ₂]
|
||||
ring
|
||||
intro k _ hkj
|
||||
erw [basis!_on_δ!₁_other hkj.symm, basis!_on_δ!₂_other hkj.symm]
|
||||
simp
|
||||
simp
|
||||
|
||||
|
||||
lemma P_zero (f : Fin n → ℚ) (h : P f = 0) : ∀ i, f i = 0 := by
|
||||
intro i
|
||||
erw [← P_δ₁ f]
|
||||
rw [h]
|
||||
rfl
|
||||
|
||||
lemma P!_zero (f : Fin n → ℚ) (h : P! f = 0) : ∀ i, f i = 0 := by
|
||||
intro i
|
||||
rw [← P!_δ!₁ f]
|
||||
rw [h]
|
||||
rfl
|
||||
|
||||
lemma Pa_zero (f g : Fin n.succ → ℚ) (h : Pa f g = 0) :
|
||||
∀ i, f i = 0 := by
|
||||
have h₃ := Pa_δa₁ f g
|
||||
rw [h] at h₃
|
||||
change 0 = _ at h₃
|
||||
simp at h₃
|
||||
intro i
|
||||
have hinduc (iv : ℕ) (hiv : iv < n.succ) : f ⟨iv, hiv⟩ = 0 := by
|
||||
induction iv
|
||||
exact h₃.symm
|
||||
rename_i iv hi
|
||||
have hivi : iv < n.succ := by omega
|
||||
have hi2 := hi hivi
|
||||
have h1 := Pa_δa₄ f g ⟨iv, hivi⟩
|
||||
rw [h, hi2] at h1
|
||||
change 0 = _ at h1
|
||||
simp at h1
|
||||
have h2 := Pa_δa₂ f g ⟨iv, by omega⟩
|
||||
simp [h, h1] at h2
|
||||
exact h2.symm
|
||||
exact hinduc i.val i.prop
|
||||
|
||||
lemma Pa_zero! (f g : Fin n.succ → ℚ) (h : Pa f g = 0) :
|
||||
∀ i, g i = 0 := by
|
||||
have hf := Pa_zero f g h
|
||||
rw [Pa, P] at h
|
||||
simp [hf] at h
|
||||
exact P!_zero g h
|
||||
|
||||
/-- A point in the span of the first part of the basis. -/
|
||||
def P' (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).LinSols := ∑ i, f i • basis i
|
||||
|
||||
/-- A point in the span of the second part of the basis. -/
|
||||
def P!' (f : Fin n → ℚ) : (PureU1 (2 * n + 1)).LinSols := ∑ i, f i • basis! i
|
||||
|
||||
/-- A point in the span of the whole basis. -/
|
||||
def Pa' (f : (Fin n) ⊕ (Fin n) → ℚ) : (PureU1 (2 * n + 1)).LinSols :=
|
||||
∑ i, f i • basisa i
|
||||
|
||||
lemma Pa'_P'_P!' (f : (Fin n) ⊕ (Fin n) → ℚ) :
|
||||
Pa' f = P' (f ∘ Sum.inl) + P!' (f ∘ Sum.inr):= by
|
||||
exact Fintype.sum_sum_type _
|
||||
|
||||
lemma P'_val (f : Fin n → ℚ) : (P' f).val = P f := by
|
||||
simp [P', P]
|
||||
funext i
|
||||
rw [sum_of_anomaly_free_linear, sum_of_charges]
|
||||
simp [HSMul.hSMul]
|
||||
|
||||
lemma P!'_val (f : Fin n → ℚ) : (P!' f).val = P! f := by
|
||||
simp [P!', P!]
|
||||
funext i
|
||||
rw [sum_of_anomaly_free_linear, sum_of_charges]
|
||||
simp [HSMul.hSMul]
|
||||
|
||||
theorem basis_linear_independent : LinearIndependent ℚ (@basis n) := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
change P' f = 0 at h
|
||||
have h1 : (P' f).val = 0 := by
|
||||
simp [h]
|
||||
rfl
|
||||
rw [P'_val] at h1
|
||||
exact P_zero f h1
|
||||
|
||||
theorem basis!_linear_independent : LinearIndependent ℚ (@basis! n) := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
change P!' f = 0 at h
|
||||
have h1 : (P!' f).val = 0 := by
|
||||
simp [h]
|
||||
rfl
|
||||
rw [P!'_val] at h1
|
||||
exact P!_zero f h1
|
||||
|
||||
|
||||
theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n.succ) := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro f h
|
||||
change Pa' f = 0 at h
|
||||
have h1 : (Pa' f).val = 0 := by
|
||||
simp [h]
|
||||
rfl
|
||||
rw [Pa'_P'_P!'] at h1
|
||||
change (P' (f ∘ Sum.inl)).val + (P!' (f ∘ Sum.inr)).val = 0 at h1
|
||||
rw [P!'_val, P'_val] at h1
|
||||
change Pa (f ∘ Sum.inl) (f ∘ Sum.inr) = 0 at h1
|
||||
have hf := Pa_zero (f ∘ Sum.inl) (f ∘ Sum.inr) h1
|
||||
have hg := Pa_zero! (f ∘ Sum.inl) (f ∘ Sum.inr) h1
|
||||
intro i
|
||||
simp_all
|
||||
cases i
|
||||
simp_all
|
||||
simp_all
|
||||
|
||||
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n.succ) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
funext i
|
||||
rw [Pa', Pa'] at h
|
||||
have h1 : ∑ i : Fin n.succ ⊕ Fin n.succ, (f i + (- f' i)) • basisa i = 0 := by
|
||||
simp only [add_smul, neg_smul]
|
||||
rw [Finset.sum_add_distrib]
|
||||
rw [h]
|
||||
rw [← Finset.sum_add_distrib]
|
||||
simp
|
||||
have h2 : ∀ i, (f i + (- f' i)) = 0 := by
|
||||
exact Fintype.linearIndependent_iff.mp (@basisa_linear_independent (n))
|
||||
(fun i => f i + -f' i) h1
|
||||
have h2i := h2 i
|
||||
linarith
|
||||
intro h
|
||||
rw [h]
|
||||
|
||||
/-- A helper function for what follows. TODO: replace this with mathlib functions. -/
|
||||
def join (g f : Fin n → ℚ) : Fin n ⊕ Fin n → ℚ := fun i =>
|
||||
match i with
|
||||
| .inl i => g i
|
||||
| .inr i => f i
|
||||
|
||||
lemma join_ext (g g' : Fin n → ℚ) (f f' : Fin n → ℚ) :
|
||||
join g f = join g' f' ↔ g = g' ∧ f = f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
apply And.intro
|
||||
funext i
|
||||
exact congr_fun h (Sum.inl i)
|
||||
funext i
|
||||
exact congr_fun h (Sum.inr i)
|
||||
intro h
|
||||
rw [h.left, h.right]
|
||||
|
||||
lemma join_Pa (g g' : Fin n.succ → ℚ) (f f' : Fin n.succ → ℚ) :
|
||||
Pa' (join g f) = Pa' (join g' f') ↔ Pa g f = Pa g' f' := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [Pa'_eq] at h
|
||||
rw [join_ext] at h
|
||||
rw [h.left, h.right]
|
||||
intro h
|
||||
apply ACCSystemLinear.LinSols.ext
|
||||
rw [Pa'_P'_P!', Pa'_P'_P!']
|
||||
simp [P'_val, P!'_val]
|
||||
exact h
|
||||
|
||||
lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n.succ → ℚ) :
|
||||
Pa g f = Pa g' f' ↔ g = g' ∧ f = f' := by
|
||||
rw [← join_Pa]
|
||||
rw [← join_ext]
|
||||
exact Pa'_eq _ _
|
||||
|
||||
|
||||
|
||||
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n.succ)) =
|
||||
FiniteDimensional.finrank ℚ (PureU1 (2 * n.succ + 1)).LinSols := by
|
||||
erw [BasisLinear.finrank_AnomalyFreeLinear]
|
||||
simp
|
||||
omega
|
||||
|
||||
/-- The basis formed out of our basisa vectors. -/
|
||||
noncomputable def basisaAsBasis :
|
||||
Basis (Fin n.succ ⊕ Fin n.succ) ℚ (PureU1 (2 * n.succ + 1)).LinSols :=
|
||||
basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card
|
||||
|
||||
lemma span_basis (S : (PureU1 (2 * n.succ + 1)).LinSols) :
|
||||
∃ (g f : Fin n.succ → ℚ) , S.val = P g + P! f := by
|
||||
have h := (mem_span_range_iff_exists_fun ℚ).mp (Basis.mem_span basisaAsBasis S)
|
||||
obtain ⟨f, hf⟩ := h
|
||||
simp [basisaAsBasis] at hf
|
||||
change P' _ + P!' _ = S at hf
|
||||
use f ∘ Sum.inl
|
||||
use f ∘ Sum.inr
|
||||
rw [← hf]
|
||||
simp [P'_val, P!'_val]
|
||||
rfl
|
||||
|
||||
lemma span_basis_swap! {S : (PureU1 (2 * n.succ + 1)).LinSols} (j : Fin n.succ)
|
||||
(hS : ((FamilyPermutations (2 * n.succ + 1)).linSolRep
|
||||
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') (g f : Fin n.succ → ℚ) (hS1 : S.val = P g + P! f):
|
||||
∃ (g' f' : Fin n.succ → ℚ),
|
||||
S'.val = P g' + P! f' ∧ P! f' = P! f +
|
||||
(S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j ∧ g' = g := by
|
||||
let X := P! f + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j
|
||||
have hf : P! f ∈ Submodule.span ℚ (Set.range basis!AsCharges) := by
|
||||
rw [(mem_span_range_iff_exists_fun ℚ)]
|
||||
use f
|
||||
rfl
|
||||
have hP : (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j ∈
|
||||
Submodule.span ℚ (Set.range basis!AsCharges) := by
|
||||
apply Submodule.smul_mem
|
||||
apply SetLike.mem_of_subset
|
||||
apply Submodule.subset_span
|
||||
simp_all only [Set.mem_range, exists_apply_eq_apply]
|
||||
have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
|
||||
apply Submodule.add_mem
|
||||
exact hf
|
||||
exact hP
|
||||
have hXsum := (mem_span_range_iff_exists_fun ℚ).mp hX
|
||||
obtain ⟨f', hf'⟩ := hXsum
|
||||
use g
|
||||
use f'
|
||||
change P! f' = _ at hf'
|
||||
erw [hf']
|
||||
simp
|
||||
change S'.val = P g + (P! f + _)
|
||||
rw [← add_assoc, ← hS1]
|
||||
apply swap!_as_add at hS
|
||||
exact hS
|
||||
|
||||
|
||||
end VectorLikeOddPlane
|
||||
|
||||
|
||||
end PureU1
|
||||
Loading…
Add table
Add a link
Reference in a new issue