feat: Add low dim U(1)

HepLean/AnomalyCancellation/PureU1/LowDim/Two.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.AnomalyCancellation.PureU1.Permutations
+/-!
+# The Pure U(1) case with 2 fermions
+
+We define an equivalence between `LinSols` and `Sols`.
+-/
+
+universe v u
+
+open Nat
+open  Finset
+
+namespace PureU1
+
+variable {n : ℕ}
+
+namespace Two
+
+/-- An equivalence between `LinSols` and `Sols`. -/
+def equiv : (PureU1 2).LinSols ≃ (PureU1 2).Sols where
+  toFun S := ⟨⟨S, by intro i; simp at i; exact Fin.elim0 i⟩, by
+    have hLin := pureU1_linear S
+    simp at hLin
+    erw [accCube_explicit]
+    simp
+    rw [show S.val (0 : Fin 2) = - S.val (1 : Fin 2) by linear_combination hLin]
+    ring⟩
+  invFun S := S.1.1
+  left_inv S := rfl
+  right_inv S := rfl
+
+end Two
+
+end PureU1