feat: Add theorems related to Sols

HepLean/AnomalyCancellation/SMNu/PlusU1/HyperCharge.lean

@@ -0,0 +1,146 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.AnomalyCancellation.SMNu.PlusU1.Basic
+import HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps
+/-!
+# Hypercharge in SM with RHN.
+
+Relavent definitions for the SM hypercharge.
+
+-/
+universe v u
+
+namespace SMRHN
+namespace PlusU1
+
+open SMνCharges
+open SMνACCs
+open BigOperators
+
+/-- The hypercharge for 1 family. -/
+@[simps!]
+def Y₁ : (PlusU1 1).Sols where
+  val := fun i =>
+    match i with
+    | (0 : Fin 6) => 1
+    | (1 : Fin 6)  => -4
+    | (2 : Fin 6)  => 2
+    | (3 : Fin 6)  => -3
+    | (4 : Fin 6)  => 6
+    | (5 : Fin 6)  => 0
+  linearSol := by
+    intro i
+    simp at i
+    match i with
+    | 0 => rfl
+    | 1 => rfl
+    | 2 => rfl
+    | 3 => rfl
+  quadSol := by
+    intro i
+    simp at i
+    match i with
+    | 0 => rfl
+  cubicSol := by rfl
+
+/-- The hypercharge for `n` family. -/
+@[simps!]
+def Y (n : ℕ) : (PlusU1 n).Sols :=
+  familyUniversalAF n Y₁
+
+namespace Y
+
+variable {n : ℕ}
+
+lemma on_quadBiLin (S : (PlusU1 n).charges) :
+    quadBiLin ((Y n).val, S) = accYY S := by
+  erw [familyUniversal_quadBiLin]
+  rw [accYY_decomp]
+  simp
+  ring_nf
+  simp
+
+lemma on_quadBiLin_AFL (S : (PlusU1 n).LinSols) : quadBiLin ((Y n).val, S.val) = 0 := by
+  rw [on_quadBiLin]
+  rw [YYsol S]
+
+
+lemma add_AFL_quad (S : (PlusU1 n).LinSols) (a b : ℚ) :
+    accQuad (a • S.val + b • (Y n).val) = a ^ 2 * accQuad S.val := by
+  erw [BiLinearSymm.toHomogeneousQuad_add, quadSol (b • (Y n)).1]
+  rw [quadBiLin.map_smul₁, quadBiLin.map_smul₂, quadBiLin.swap, on_quadBiLin_AFL]
+  erw [accQuad.map_smul]
+  simp
+
+lemma add_quad (S : (PlusU1 n).QuadSols) (a b : ℚ) :
+    accQuad (a • S.val + b • (Y n).val) = 0 := by
+  rw [add_AFL_quad, quadSol S]
+  simp
+
+/-- The `QuadSol` obtained by adding hypercharge to a `QuadSol`. -/
+def addQuad (S : (PlusU1 n).QuadSols) (a b : ℚ) : (PlusU1 n).QuadSols :=
+  linearToQuad (a • S.1 + b • (Y n).1.1) (add_quad S a b)
+
+lemma addQuad_zero (S : (PlusU1 n).QuadSols) (a : ℚ): addQuad S a 0 = a • S := by
+  simp [addQuad, linearToQuad]
+  rfl
+
+lemma on_cubeTriLin (S : (PlusU1 n).charges) :
+    cubeTriLin ((Y n).val, (Y n).val, S) =  6 * accYY S := by
+  erw [familyUniversal_cubeTriLin']
+  rw [accYY_decomp]
+  simp
+  ring
+
+lemma on_cubeTriLin_AFL (S : (PlusU1 n).LinSols) :
+    cubeTriLin ((Y n).val, (Y n).val, S.val) = 0 := by
+  rw [on_cubeTriLin]
+  rw [YYsol S]
+  simp
+
+lemma on_cubeTriLin' (S : (PlusU1 n).charges) :
+    cubeTriLin ((Y n).val, S, S) =  6 * accQuad S := by
+  erw [familyUniversal_cubeTriLin]
+  rw [accQuad_decomp]
+  simp
+  ring_nf
+
+lemma on_cubeTriLin'_ALQ (S : (PlusU1 n).QuadSols) :
+    cubeTriLin ((Y n).val, S.val, S.val) = 0 := by
+  rw [on_cubeTriLin']
+  rw [quadSol S]
+  simp
+
+lemma add_AFL_cube (S : (PlusU1 n).LinSols) (a b : ℚ) :
+    accCube (a • S.val + b • (Y n).val) =
+    a ^ 2 * (a * accCube S.val + 3 * b * cubeTriLin (S.val, S.val, (Y n).val)) := by
+  erw [TriLinearSymm.toCubic_add, cubeSol (b • (Y n)), accCube.map_smul]
+  repeat rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
+  rw [on_cubeTriLin_AFL]
+  simp
+  ring
+
+lemma add_AFQ_cube (S : (PlusU1 n).QuadSols) (a b : ℚ) :
+    accCube (a • S.val + b • (Y n).val) = a ^ 3 *  accCube S.val := by
+  rw [add_AFL_cube]
+  rw [cubeTriLin.swap₃]
+  rw [on_cubeTriLin'_ALQ]
+  ring
+
+lemma add_AF_cube (S : (PlusU1 n).Sols) (a b : ℚ) :
+    accCube (a • S.val + b • (Y n).val) = 0 := by
+  rw [add_AFQ_cube]
+  rw [cubeSol S]
+  simp
+
+/-- The `Sol` obtained by adding hypercharge to a `Sol`. -/
+def addCube (S : (PlusU1 n).Sols) (a b : ℚ) : (PlusU1 n).Sols :=
+  quadToAF (addQuad S.1 a b) (add_AF_cube S a b)
+
+
+end Y
+end PlusU1
+end SMRHN