/- Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved. Released under Apache 2.0 license. Authors: Joseph Tooby-Smith -/ import HepLean.AnomalyCancellation.MSSMNu.Basic import HepLean.AnomalyCancellation.MSSMNu.LineY3B3 import Mathlib.Tactic.Polyrith import Mathlib.Tactic.Linarith /-! # The type of solutions perpendicular to `Y₃` and `B₃` We define the type of solutions which are orthogonal to `Y₃` and `B₃` and prove some basic lemmas about them. # References The main reference for the material in this file is: - https://arxiv.org/pdf/2107.07926.pdf -/ universe v u namespace MSSMACC open MSSMCharges open MSSMACCs open BigOperators /-- The type of linear solutions orthogonal to $Y_3$ and $B_3$. -/ structure AnomalyFreePerp extends MSSMACC.LinSols where perpY₃ : dot Y₃.val val = 0 perpB₃ : dot B₃.val val = 0 /-- The projection of an object in `MSSMACC.AnomalyFreeLinear` onto the subspace orthgonal to `Y₃` and`B₃`. -/ def proj (T : MSSMACC.LinSols) : MSSMACC.AnomalyFreePerp := ⟨(dot B₃.val T.val - dot Y₃.val T.val) • Y₃.1.1 + (dot Y₃.val T.val - 2 * dot B₃.val T.val) • B₃.1.1 + dot Y₃.val B₃.val • T, by change dot _ (_ • Y₃.val + _ • B₃.val + _ • T.val) = 0 rw [dot.map_add₂, dot.map_add₂] rw [dot.map_smul₂, dot.map_smul₂, dot.map_smul₂] rw [show dot Y₃.val B₃.val = 108 by rfl] rw [show dot Y₃.val Y₃.val = 216 by rfl] ring, by change dot _ (_ • Y₃.val + _ • B₃.val + _ • T.val) = 0 rw [dot.map_add₂, dot.map_add₂] rw [dot.map_smul₂, dot.map_smul₂, dot.map_smul₂] rw [show dot Y₃.val B₃.val = 108 by rfl] rw [show dot B₃.val Y₃.val = 108 by rfl] rw [show dot B₃.val B₃.val = 108 by rfl] ring⟩ lemma proj_val (T : MSSMACC.LinSols) : (proj T).val = (dot B₃.val T.val - dot Y₃.val T.val) • Y₃.val + ( (dot Y₃.val T.val - 2 * dot B₃.val T.val)) • B₃.val + dot Y₃.val B₃.val • T.val := by rfl lemma Y₃_plus_B₃_plus_proj (T : MSSMACC.LinSols) (a b c : ℚ) : a • Y₃.val + b • B₃.val + c • (proj T).val = (a + c * (dot B₃.val T.val - dot Y₃.val T.val)) • Y₃.val + (b + c * (dot Y₃.val T.val - 2 * dot B₃.val T.val)) • B₃.val + (dot Y₃.val B₃.val * c) • T.val:= by rw [proj_val] rw [DistribMulAction.smul_add, DistribMulAction.smul_add] rw [add_assoc (_ • _ • Y₃.val), ← add_assoc (_ • Y₃.val + _ • B₃.val), add_assoc (_ • Y₃.val)] rw [add_comm (_ • B₃.val) (_ • _ • Y₃.val), ← add_assoc (_ • Y₃.val)] rw [← MulAction.mul_smul, ← Module.add_smul] repeat rw [add_assoc] apply congrArg rw [← add_assoc, ← MulAction.mul_smul, ← Module.add_smul] apply congrArg simp only [HSMul.hSMul, SMul.smul, MSSMACC_numberCharges, Fin.isValue, Fin.reduceFinMk] funext i linarith lemma quad_Y₃_proj (T : MSSMACC.LinSols) : quadBiLin Y₃.val (proj T).val = dot Y₃.val B₃.val * quadBiLin Y₃.val T.val := by rw [proj_val] rw [quadBiLin.map_add₂, quadBiLin.map_add₂] rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂] rw [show quadBiLin Y₃.val B₃.val = 0 by rfl] rw [show quadBiLin Y₃.val Y₃.val = 0 by rfl] ring lemma quad_B₃_proj (T : MSSMACC.LinSols) : quadBiLin B₃.val (proj T).val = dot Y₃.val B₃.val * quadBiLin B₃.val T.val := by rw [proj_val] rw [quadBiLin.map_add₂, quadBiLin.map_add₂] rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂] rw [show quadBiLin B₃.val Y₃.val = 0 by rfl] rw [show quadBiLin B₃.val B₃.val = 0 by rfl] ring lemma quad_self_proj (T : MSSMACC.Sols) : quadBiLin T.val (proj T.1.1).val = (dot B₃.val T.val - dot Y₃.val T.val) * quadBiLin Y₃.val T.val + (dot Y₃.val T.val - 2 * dot B₃.val T.val) * quadBiLin B₃.val T.val := by rw [proj_val] rw [quadBiLin.map_add₂, quadBiLin.map_add₂] rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂] erw [quadSol T.1] rw [quadBiLin.swap T.val Y₃.val, quadBiLin.swap T.val B₃.val] ring lemma quad_proj (T : MSSMACC.Sols) : quadBiLin (proj T.1.1).val (proj T.1.1).val = 2 * dot Y₃.val B₃.val * ((dot B₃.val T.val - dot Y₃.val T.val) * quadBiLin Y₃.val T.val + (dot Y₃.val T.val - 2 * dot B₃.val T.val) * quadBiLin B₃.val T.val ) := by nth_rewrite 1 [proj_val] repeat rw [quadBiLin.map_add₁] repeat rw [quadBiLin.map_smul₁] rw [quad_Y₃_proj, quad_B₃_proj, quad_self_proj] ring lemma cube_proj_proj_Y₃ (T : MSSMACC.LinSols) : cubeTriLin (proj T).val (proj T).val Y₃.val = (dot Y₃.val B₃.val)^2 * cubeTriLin T.val T.val Y₃.val := by rw [proj_val] rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂] erw [lineY₃B₃_doublePoint] rw [cubeTriLin.map_add₂] rw [cubeTriLin.swap₂] rw [cubeTriLin.map_add₁, cubeTriLin.map_smul₁, cubeTriLin.map_smul₃] rw [doublePoint_Y₃_Y₃] rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₃, cubeTriLin.swap₁] rw [doublePoint_Y₃_B₃] rw [cubeTriLin.map_add₂] rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂] rw [cubeTriLin.swap₁, cubeTriLin.swap₂] rw [doublePoint_Y₃_Y₃] rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂] rw [cubeTriLin.swap₁, cubeTriLin.swap₂, cubeTriLin.swap₁] rw [doublePoint_Y₃_B₃] rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂] ring lemma cube_proj_proj_B₃ (T : MSSMACC.LinSols) : cubeTriLin (proj T).val (proj T).val B₃.val = (dot Y₃.val B₃.val)^2 * cubeTriLin T.val T.val B₃.val := by rw [proj_val] rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂] erw [lineY₃B₃_doublePoint] rw [cubeTriLin.map_add₂, cubeTriLin.swap₂, cubeTriLin.map_add₁, cubeTriLin.map_smul₁, cubeTriLin.map_smul₃, doublePoint_Y₃_B₃] rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₃, cubeTriLin.swap₁, doublePoint_B₃_B₃] rw [cubeTriLin.map_add₂, cubeTriLin.map_smul₁, cubeTriLin.map_smul₂] rw [cubeTriLin.swap₁, cubeTriLin.swap₂, doublePoint_Y₃_B₃] rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.swap₁, cubeTriLin.swap₂, cubeTriLin.swap₁, doublePoint_B₃_B₃] rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂] ring lemma cube_proj_proj_self (T : MSSMACC.Sols) : cubeTriLin (proj T.1.1).val (proj T.1.1).val T.val = 2 * dot Y₃.val B₃.val * ((dot B₃.val T.val - dot Y₃.val T.val) * cubeTriLin T.val T.val Y₃.val + ( dot Y₃.val T.val- 2 * dot B₃.val T.val) * cubeTriLin T.val T.val B₃.val) := by rw [proj_val] rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂] erw [lineY₃B₃_doublePoint] repeat rw [cubeTriLin.map_add₁] repeat rw [cubeTriLin.map_smul₁] repeat rw [cubeTriLin.map_add₂] repeat rw [cubeTriLin.map_smul₂] erw [T.cubicSol] rw [cubeTriLin.swap₁ Y₃.val T.val T.val, cubeTriLin.swap₂ T.val Y₃.val T.val] rw [cubeTriLin.swap₁ B₃.val T.val T.val, cubeTriLin.swap₂ T.val B₃.val T.val] ring lemma cube_proj (T : MSSMACC.Sols) : cubeTriLin (proj T.1.1).val (proj T.1.1).val (proj T.1.1).val = 3 * dot Y₃.val B₃.val ^ 2 * ((dot B₃.val T.val - dot Y₃.val T.val) * cubeTriLin T.val T.val Y₃.val + (dot Y₃.val T.val - 2 * dot B₃.val T.val) * cubeTriLin T.val T.val B₃.val) := by nth_rewrite 3 [proj_val] repeat rw [cubeTriLin.map_add₃] repeat rw [cubeTriLin.map_smul₃] rw [cube_proj_proj_Y₃, cube_proj_proj_B₃, cube_proj_proj_self] ring end MSSMACC