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.Sort
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import Mathlib.Logic.Equiv.Fin
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/-!
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# Vector like charges
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For the `n`-even case we define the property of a charge assignment being vector like.
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## TODO
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The `n`-odd case.
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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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/--
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Given a natural number `n`, this lemma proves that `n + n` is equal to `2 * n`.
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-/
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lemma split_equal (n : ℕ) : n + n = 2 * n := (Nat.two_mul n).symm
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lemma split_odd (n : ℕ) : n + 1 + n = 2 * n + 1 := by omega
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/-- A charge configuration for n even is vector like if when sorted the `i`th element
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is equal to the negative of the `n + i`th element. -/
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@[simp]
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2024-06-26 11:54:02 -04:00
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def VectorLikeEven (S : (PureU1 (2 * n)).Charges) : Prop :=
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2024-04-18 10:09:08 -04:00
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∀ (i : Fin n), (sort S) (Fin.cast (split_equal n) (Fin.castAdd n i))
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= - (sort S) (Fin.cast (split_equal n) (Fin.natAdd n i))
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end PureU1
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