174 lines
7.2 KiB
Text
174 lines
7.2 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.PureU1.Basic
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import HepLean.AnomalyCancellation.PureU1.ConstAbs
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import HepLean.AnomalyCancellation.PureU1.Even.BasisLinear
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import HepLean.AnomalyCancellation.PureU1.LineInPlaneCond
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import HepLean.AnomalyCancellation.PureU1.Permutations
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import Mathlib.RepresentationTheory.Basic
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import Mathlib.Tactic.Polyrith
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/-!
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# Line In Cubic Even case
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We say that a linear solution satisfies the `lineInCubic` property
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if the line through that point and through the two different planes formed by the basis of
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`LinSols` lies in the cubic.
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We show that for a solution all its permutations satisfy this property, then there exists
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a permutation for which it lies in the plane spanned by the first part of the basis.
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The main reference for this file is:
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- https://arxiv.org/pdf/1912.04804.pdf
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-/
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namespace PureU1
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namespace Even
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open BigOperators
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variable {n : ℕ}
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open VectorLikeEvenPlane
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/-- A property on `LinSols`, satisfied if every point on the line between the two planes
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in the basis through that point is in the cubic. -/
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def LineInCubic (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
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∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ),
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accCube (2 * n.succ) (a • P g + b • P! f) = 0
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lemma lineInCubic_expand {S : (PureU1 (2 * n.succ)).LinSols} (h : LineInCubic S) :
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∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ),
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3 * a * b * (a * accCubeTriLinSymm (P g) (P g) (P! f)
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+ b * accCubeTriLinSymm (P! f) (P! f) (P g)) = 0 := by
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intro g f hS a b
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have h1 := h g f hS a b
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change accCubeTriLinSymm.toCubic (a • P g + b • P! f) = 0 at h1
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simp only [TriLinearSymm.toCubic_add] at h1
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simp only [HomogeneousCubic.map_smul,
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accCubeTriLinSymm.map_smul₁, accCubeTriLinSymm.map_smul₂, accCubeTriLinSymm.map_smul₃] at h1
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erw [P_accCube, P!_accCube] at h1
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rw [← h1]
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ring
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/--
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This lemma states that for a given `S` of type `(PureU1 (2 * n.succ)).AnomalyFreeLinear` and
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a proof `h` that the line through `S` lies on a cubic curve,
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for any functions `g : Fin n.succ → ℚ` and `f : Fin n → ℚ`, if `S.val = P g + P! f`,
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then `accCubeTriLinSymm.toFun (P g, P g, P! f) = 0`.
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-/
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lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n.succ)).LinSols} (h : LineInCubic S) :
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∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f),
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accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
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intro g f hS
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linear_combination 2 / 3 * (lineInCubic_expand h g f hS 1 1) -
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(lineInCubic_expand h g f hS 1 2) / 6
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/-- A `LinSol` satisfies `LineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
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def LineInCubicPerm (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
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∀ (M : (FamilyPermutations (2 * n.succ)).group),
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LineInCubic ((FamilyPermutations (2 * n.succ)).linSolRep M S)
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/-- If `lineInCubicPerm S` then `lineInCubic S`. -/
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lemma lineInCubicPerm_self {S : (PureU1 (2 * n.succ)).LinSols}
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(hS : LineInCubicPerm S) : LineInCubic S := hS 1
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/-- If `lineInCubicPerm S` then `lineInCubicPerm (M S)` for all permutations `M`. -/
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lemma lineInCubicPerm_permute {S : (PureU1 (2 * n.succ)).LinSols}
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(hS : LineInCubicPerm S) (M' : (FamilyPermutations (2 * n.succ)).group) :
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LineInCubicPerm ((FamilyPermutations (2 * n.succ)).linSolRep M' S) := by
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rw [LineInCubicPerm]
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intro M
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change LineInCubic
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(((FamilyPermutations (2 * n.succ)).linSolRep M *
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(FamilyPermutations (2 * n.succ)).linSolRep M') S)
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erw [← (FamilyPermutations (2 * n.succ)).linSolRep.map_mul M M']
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exact hS (M * M')
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lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ)).LinSols}
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(LIC : LineInCubicPerm S) :
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∀ (j : Fin n) (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = Pa g f),
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(S.val (δ!₂ j) - S.val (δ!₁ j))
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* accCubeTriLinSymm (P g) (P g) (basis!AsCharges j) = 0 := by
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intro j g f h
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let S' := (FamilyPermutations (2 * n.succ)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j)) S
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have hSS' : ((FamilyPermutations (2 * n.succ)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j))) S = S' := rfl
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obtain ⟨g', f', hall⟩ := span_basis_swap! j hSS' g f h
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have h1 := line_in_cubic_P_P_P! (lineInCubicPerm_self LIC) g f h
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have h2 := line_in_cubic_P_P_P!
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(lineInCubicPerm_self (lineInCubicPerm_permute LIC (pairSwap (δ!₁ j) (δ!₂ j)))) g' f' hall.1
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rw [hall.2.1, hall.2.2] at h2
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rw [accCubeTriLinSymm.map_add₃, h1, accCubeTriLinSymm.map_smul₃] at h2
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simpa using h2
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lemma P_P_P!_accCube' {S : (PureU1 (2 * n.succ.succ)).LinSols}
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(f : Fin n.succ.succ → ℚ) (g : Fin n.succ → ℚ) (hS : S.val = Pa f g) :
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accCubeTriLinSymm (P f) (P f) (basis!AsCharges (Fin.last n)) =
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- (S.val (δ!₂ (Fin.last n)) + S.val (δ!₁ (Fin.last n))) * (2 * S.val δ!₄ +
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S.val (δ!₂ (Fin.last n)) + S.val (δ!₁ (Fin.last n))) := by
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rw [P_P_P!_accCube f (Fin.last n)]
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have h1 := Pa_δ!₄ f g
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have h2 := Pa_δ!₁ f g (Fin.last n)
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have h3 := Pa_δ!₂ f g (Fin.last n)
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simp at h1 h2 h3
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have hl : f (Fin.succ (Fin.last n)) = - Pa f g δ!₄ := by
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simp_all only [Fin.succ_last, neg_neg]
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erw [hl] at h2
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have hg : g (Fin.last n) = Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄ := by
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linear_combination -(1 * h2)
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have hll : f (Fin.castSucc (Fin.last n)) =
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- (Pa f g (δ!₂ (Fin.last n)) + Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄) := by
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linear_combination h3 - 1 * hg
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rw [← hS] at hl hll
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rw [hl, hll]
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ring
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lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ)).LinSols}
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(LIC : LineInCubicPerm S) :
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LineInPlaneProp
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((S.val (δ!₂ (Fin.last n))), ((S.val (δ!₁ (Fin.last n))), (S.val δ!₄))) := by
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obtain ⟨g, f, hfg⟩ := span_basis S
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have h1 := lineInCubicPerm_swap LIC (Fin.last n) g f hfg
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rw [P_P_P!_accCube' g f hfg] at h1
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simp at h1
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cases h1 <;> rename_i h1
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apply Or.inl
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linear_combination h1
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cases h1 <;> rename_i h1
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apply Or.inr
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apply Or.inl
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linear_combination -(1 * h1)
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apply Or.inr
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apply Or.inr
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exact h1
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lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
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(LIC : LineInCubicPerm S) : LineInPlaneCond S := by
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refine @Prop_three (2 * n.succ.succ) LineInPlaneProp S (δ!₂ (Fin.last n)) (δ!₁ (Fin.last n))
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δ!₄ ?_ ?_ ?_ ?_
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simp [Fin.ext_iff, δ!₂, δ!₁]
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simp [Fin.ext_iff, δ!₂, δ!₄]
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simp [Fin.ext_iff, δ!₁, δ!₄]
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omega
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intro M
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exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
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lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ)).Sols}
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(LIC : LineInCubicPerm S.1.1) : ConstAbs S.val :=
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linesInPlane_constAbs_AF S (lineInCubicPerm_last_perm LIC)
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theorem lineInCubicPerm_vectorLike {S : (PureU1 (2 * n.succ.succ)).Sols}
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(LIC : LineInCubicPerm S.1.1) : VectorLikeEven S.val :=
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ConstAbs.boundary_value_even S.1.1 (lineInCubicPerm_constAbs LIC)
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theorem lineInCubicPerm_in_plane (S : (PureU1 (2 * n.succ.succ)).Sols)
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(LIC : LineInCubicPerm S.1.1) : ∃ (M : (FamilyPermutations (2 * n.succ.succ)).group),
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(FamilyPermutations (2 * n.succ.succ)).linSolRep M S.1.1
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∈ Submodule.span ℚ (Set.range basis) :=
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vectorLikeEven_in_span S.1.1 (lineInCubicPerm_vectorLike LIC)
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end Even
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end PureU1
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