| import OAI.Analysis.HyperbolicCones.MatrixPositive |
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| namespace OAI |
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| noncomputable section |
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| open Set Filter Matrix |
| open scoped Topology Matrix.Norms.L2Operator MatrixOrder |
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| namespace Paper256 |
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| def symInverse {n : ℕ} (X : Sym n) : Sym n := |
| ⟨(X : Mat n ℝ)⁻¹, Matrix.IsHermitian.inv X.property⟩ |
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| def phiSym (y : Fin 3 → ℝ) (X : Sym 4) : Sym 4 := |
| ⟨phi y (X : Mat 4 ℝ), phi_isHermitian y _ X.property⟩ |
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| def coneResidual (X Z : Sym 4) (y : Fin 3 → ℝ) (t : ℝ) : Sym 4 := |
| Z + t • 1 - phiSym y (symInverse (X + t • 1)) |
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| theorem continuousAt_symInverse {n : ℕ} (X : Sym n) |
| (hX : IsUnit (X : Mat n ℝ)) : ContinuousAt symInverse X := by |
| have hd : (X : Mat n ℝ).det ≠ 0 := |
| ((Matrix.isUnit_iff_isUnit_det _).mp hX).ne_zero |
| have hi : ContinuousAt Inv.inv (X : Mat n ℝ) := |
| continuousAt_matrix_inv _ (by |
| simpa only [Ring.inverse_eq_inv'] using (continuousAt_inv₀ hd)) |
| apply tendsto_subtype_rng.mpr |
| exact hi.comp (f := fun H : Sym n => (H : Mat n ℝ)) (x := X) |
| continuous_subtype_val.continuousAt |
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| theorem continuous_phiSym (y : Fin 3 → ℝ) : Continuous (phiSym y) := by |
| exact ((phiLinear y).continuous_of_finiteDimensional.comp continuous_subtype_val).subtype_mk _ |
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| theorem positive_shift {n : ℕ} (X : Sym n) (hX : (X : Mat n ℝ).PosDef) |
| (t : ℝ) (ht : 0 ≤ t) : ((X + t • 1 : Sym n) : Mat n ℝ).PosDef := |
| hX.add_posSemidef (Matrix.PosSemidef.one.smul ht) |
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| theorem continuousAt_coneResidual (X Z : Sym 4) (y : Fin 3 → ℝ) (t : ℝ) |
| (ht : ((X + t • 1 : Sym 4) : Mat 4 ℝ).PosDef) : |
| ContinuousAt (coneResidual X Z y) t := by |
| have hX : Continuous (fun s : ℝ => X + s • (1 : Sym 4)) := by fun_prop |
| have hZ : Continuous (fun s : ℝ => Z + s • (1 : Sym 4)) := by fun_prop |
| have hi : ContinuousAt (fun s : ℝ => symInverse (X + s • (1 : Sym 4))) t := |
| (continuousAt_symInverse _ ht.isUnit).comp |
| (f := fun s : ℝ => X + s • (1 : Sym 4)) hX.continuousAt |
| exact hZ.continuousAt.sub ((continuous_phiSym y).continuousAt.comp hi) |
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| theorem continuousOn_coneResidual (X Z : Sym 4) (y : Fin 3 → ℝ) |
| (hX : (X : Mat 4 ℝ).PosDef) : ContinuousOn (coneResidual X Z y) (Ici 0) := by |
| intro t ht |
| exact (continuousAt_coneResidual X Z y t (positive_shift X hX t ht)).continuousWithinAt |
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| end Paper256 |
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| end |
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| end OAI |
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