| import OAI.Geometry.RuledCovering.Main |
| import OAI.Geometry.CylinderCovering.Ruled.ZeroDiagonalDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| namespace RuledApproximation |
|
|
| lemma inner_vec2_left (x y : ℝ) (z : Plane) : |
| inner ℝ (vec2 x y) z = x * z 0 + y * z 1 := by |
| nth_rw 1 [← vec2_coordinates z] |
| simp [vec2, EuclideanSpace.inner_toLp_toLp, dotProduct, Fin.sum_univ_succ] |
| ring |
|
|
| lemma plane_trace_coordinates (A : Plane →L[ℝ] Plane) : |
| LinearMap.trace ℝ Plane A.toLinearMap = |
| A (vec2 1 0) 0 + A (vec2 0 1) 1 := by |
| have h0 : (EuclideanSpace.single 0 1 : Plane) = vec2 1 0 := by |
| ext i; fin_cases i <;> simp [vec2] |
| have h1 : (EuclideanSpace.single 1 1 : Plane) = vec2 0 1 := by |
| ext i; fin_cases i <;> simp [vec2] |
| rw [LinearMap.trace_eq_matrix_trace ℝ (EuclideanSpace.basisFun (Fin 2) ℝ).toBasis, |
| Matrix.trace_fin_two] |
| simp [LinearMap.toMatrix_apply, EuclideanSpace.basisFun_apply, |
| EuclideanSpace.basisFun_repr, h0, h1] |
|
|
| lemma trace_zero_unit_isotropic (A : Plane →L[ℝ] Plane) |
| (htr : LinearMap.trace ℝ Plane A.toLinearMap = 0) : |
| ∃ u : Plane, ‖u‖ = 1 ∧ inner ℝ u (A u) = 0 := by |
| let p : ℝ → Plane := fun t => vec2 (1 - t) t |
| let q : ℝ → ℝ := fun t => inner ℝ (p t) (A (p t)) |
| have hq : Continuous q := by dsimp [q, p, vec2]; fun_prop |
| have hsum : q 0 + q 1 = 0 := by |
| rw [plane_trace_coordinates] at htr |
| simpa [q, p, inner_vec2_left] using htr |
| have hex : ∃ t ∈ Icc (0 : ℝ) 1, q t = 0 := by |
| by_cases hzero : q 0 ≤ 0 |
| · exact intermediate_value_Icc (by norm_num) hq.continuousOn |
| ⟨hzero, by linarith⟩ |
| · exact intermediate_value_Icc' (by norm_num) hq.continuousOn |
| ⟨by linarith, by linarith⟩ |
| obtain ⟨t, ht, hqt⟩ := hex |
| have hne : p t ≠ 0 := by |
| intro hp |
| have h0 := congrArg (fun z : Plane => z 0) hp |
| have h1 := congrArg (fun z : Plane => z 1) hp |
| simp [p] at h0 h1 |
| linarith |
| refine ⟨‖p t‖⁻¹ • p t, ?_, ?_⟩ |
| · rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm, |
| inv_mul_cancel₀ (norm_ne_zero_iff.mpr hne)] |
| · rw [map_smul, inner_smul_right, real_inner_smul_left] |
| change ‖p t‖⁻¹ * (‖p t‖⁻¹ * q t) = 0 |
| rw [hqt] |
| ring |
|
|
| theorem orthogonalZeroDiagonal_proof : OrthogonalZeroDiagonalStatement := by |
| intro A htr hdet |
| classical |
| obtain ⟨u, hu, hAu⟩ := trace_zero_unit_isotropic A htr |
| have hon : Orthonormal ℝ (({0} : Set (Fin 2)).domRestrict (fun _ => u)) := by |
| rw [orthonormal_iff_ite] |
| intro i j |
| have hi : i = ⟨0, by simp⟩ := Subtype.ext (mem_singleton_iff.mp i.property) |
| have hj : j = ⟨0, by simp⟩ := Subtype.ext (mem_singleton_iff.mp j.property) |
| subst i |
| subst j |
| simp [hu] |
| obtain ⟨B, hB⟩ := hon.exists_orthonormalBasis_extension_of_card_eq (by simp) |
| have hB0 : B 0 = u := hB 0 (by simp) |
| let R : Plane ≃ₗᵢ[ℝ] Plane := B.repr.symm |
| let W : Plane →L[ℝ] Plane := |
| (B.repr.toContinuousLinearMap.comp A).comp B.repr.symm.toContinuousLinearMap |
| have hWconj : W.toLinearMap = B.repr.toLinearEquiv.conj A.toLinearMap := rfl |
| have he0 : vec2 1 0 = EuclideanSpace.single 0 1 := by |
| ext i; fin_cases i <;> simp [vec2] |
| have he1 : vec2 0 1 = EuclideanSpace.single 1 1 := by |
| ext i; fin_cases i <;> simp [vec2] |
| have hW00 : W (vec2 1 0) 0 = 0 := by |
| change B.repr (A (B.repr.symm (vec2 1 0))) 0 = 0 |
| rw [he0, B.repr_symm_single, B.repr_apply_apply, hB0] |
| exact hAu |
| have hWtrace : LinearMap.trace ℝ Plane W.toLinearMap = 0 := by |
| rw [hWconj, LinearMap.trace_conj'] |
| exact htr |
| have hW11 : W (vec2 0 1) 1 = 0 := by |
| rw [plane_trace_coordinates, hW00, zero_add] at hWtrace |
| exact hWtrace |
| have hexp (z : Plane) : z = z 0 • vec2 1 0 + z 1 • vec2 0 1 := by |
| ext i; fin_cases i <;> simp [vec2] |
| have hWexp (z : Plane) : |
| W z = vec2 (W (vec2 0 1) 0 * z 1) (W (vec2 1 0) 1 * z 0) := by |
| nth_rw 1 [hexp z] |
| rw [map_add, map_smul, map_smul] |
| ext i |
| fin_cases i <;> simp [hW00, hW11, mul_comm] |
| have hWmat : W = matrixCLM 0 (W (vec2 0 1) 0) (W (vec2 1 0) 1) 0 := by |
| ext z i |
| rw [hWexp] |
| simp |
| have hWdet : LinearMap.det W.toLinearMap = LinearMap.det A.toLinearMap := by |
| rw [hWconj] |
| simpa only [LinearEquiv.conj_apply, LinearMap.comp_assoc] using |
| (LinearMap.det_conj A.toLinearMap B.repr.toLinearEquiv) |
| have hprod : W (vec2 0 1) 0 * W (vec2 1 0) 1 = -LinearMap.det A.toLinearMap := by |
| rw [hWmat, matrixCLM_det] at hWdet |
| linarith |
| have hnorm (z : Plane) : ‖W z‖ ≤ ‖A‖ * ‖z‖ := by |
| change ‖B.repr (A (B.repr.symm z))‖ ≤ _ |
| rw [B.repr.norm_map] |
| exact (A.le_opNorm _).trans_eq (by rw [B.repr.symm.norm_map]) |
| refine ⟨R, W (vec2 0 1) 0, W (vec2 1 0) 1, hWexp, hprod, |
| by linarith, ?_, ?_⟩ |
| · have hb := (plane_coordinate_le (W (vec2 0 1)) 0).trans (hnorm (vec2 0 1)) |
| simpa [vec2, EuclideanSpace.norm_eq] using hb |
| · have hc := (plane_coordinate_le (W (vec2 1 0)) 1).trans (hnorm (vec2 1 0)) |
| simpa [vec2, EuclideanSpace.norm_eq] using hc |
|
|
| end RuledApproximation |
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| end |
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| end OAI |
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