2024-04-18 10:50:21 -04:00
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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2024-07-12 16:39:44 -04:00
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Released under Apache 2.0 license as described in the file LICENSE.
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2024-04-18 10:50:21 -04:00
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Authors: Joseph Tooby-Smith
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-/
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2024-06-25 07:06:32 -04:00
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import HepLean.AnomalyCancellation.PureU1.Basic
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2024-04-18 10:50:21 -04:00
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/-!
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# The Pure U(1) case with 2 fermions
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We define an equivalence between `LinSols` and `Sols`.
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-/
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universe v u
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open Nat
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2024-07-12 11:23:02 -04:00
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open Finset
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2024-04-18 10:50:21 -04:00
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namespace PureU1
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variable {n : ℕ}
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namespace Two
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/-- An equivalence between `LinSols` and `Sols`. -/
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def equiv : (PureU1 2).LinSols ≃ (PureU1 2).Sols where
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toFun S := ⟨⟨S, by intro i; simp at i; exact Fin.elim0 i⟩, by
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have hLin := pureU1_linear S
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simp at hLin
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erw [accCube_explicit]
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2024-04-18 10:50:59 -04:00
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simp only [Fin.sum_univ_two, Fin.isValue]
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2024-06-13 16:49:01 -04:00
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rw [show S.val (0 : Fin 2) = - S.val (1 : Fin 2) from eq_neg_of_add_eq_zero_left hLin]
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2024-04-18 10:50:21 -04:00
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ring⟩
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invFun S := S.1.1
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left_inv S := rfl
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right_inv S := rfl
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end Two
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end PureU1
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