2024-04-18 10:09:08 -04:00
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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.Permutations
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2024-06-25 07:06:32 -04:00
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import Mathlib.Data.Fin.Tuple.Sort
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2024-04-18 10:09:08 -04:00
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/-!
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# Sort for Pure U(1) charges
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We define the sort function for Pure U(1) charges, and prove some basic properties.
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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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namespace PureU1
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variable {n : ℕ}
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/-- We say a charge is shorted if for all `i ≤ j`, then `S i ≤ S j`. -/
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@[simp]
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def sorted {n : ℕ} (S : (PureU1 n).charges) : Prop :=
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∀ i j (_ : i ≤ j), S i ≤ S j
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/-- Given a charge assignment `S`, the corresponding sorted charge assignment. -/
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@[simp]
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def sort {n : ℕ} (S : (PureU1 n).charges) : (PureU1 n).charges :=
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((FamilyPermutations n).rep (Tuple.sort S).symm S)
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lemma sort_sorted {n : ℕ} (S : (PureU1 n).charges) : sorted (sort S) := by
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2024-04-18 10:12:55 -04:00
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simp only [sorted, PureU1_numberCharges, sort, FamilyPermutations, permGroup, permCharges,
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MonoidHom.coe_mk, OneHom.coe_mk, chargeMap_apply]
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2024-04-18 10:09:08 -04:00
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intro i j hij
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exact Tuple.monotone_sort S hij
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lemma sort_perm {n : ℕ} (S : (PureU1 n).charges) (M :(FamilyPermutations n).group) :
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sort ((FamilyPermutations n).rep M S) = sort S :=
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@Tuple.comp_perm_comp_sort_eq_comp_sort n ℚ _ S M⁻¹
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lemma sort_apply {n : ℕ} (S : (PureU1 n).charges) (j : Fin n) :
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sort S j = S ((Tuple.sort S) j) := by
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rfl
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lemma sort_zero {n : ℕ} (S : (PureU1 n).charges) (hS : sort S = 0) : S = 0 := by
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funext i
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have hj : ∀ j, sort S j = 0 := by
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rw [hS]
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intro j
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rfl
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have hi := hj ((Tuple.sort S).invFun i)
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rw [sort_apply] at hi
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simp at hi
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rw [hi]
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rfl
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lemma sort_projection {n : ℕ} (S : (PureU1 n).charges) : sort (sort S) = sort S :=
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sort_perm S (Tuple.sort S).symm
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/-- The sort function acting on `LinSols`. -/
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def sortAFL {n : ℕ} (S : (PureU1 n).LinSols) : (PureU1 n).LinSols :=
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((FamilyPermutations n).linSolRep (Tuple.sort S.val).symm S)
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lemma sortAFL_val {n : ℕ} (S : (PureU1 n).LinSols) : (sortAFL S).val = sort S.val := by
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rfl
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lemma sortAFL_zero {n : ℕ} (S : (PureU1 n).LinSols) (hS : sortAFL S = 0) : S = 0 := by
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apply ACCSystemLinear.LinSols.ext
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have h1 : sort S.val = 0 := by
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rw [← sortAFL_val]
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rw [hS]
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rfl
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exact sort_zero S.val h1
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end PureU1
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