| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
|
|
| namespace OAI |
|
|
| /-! Three-lines tangent estimates for entropy derivatives. -/ |
|
|
| noncomputable section |
| open scoped BigOperators ComplexOrder |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open scoped BigOperators |
| open scoped Topology |
| open Filter |
| open scoped MatrixOrder |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open scoped ComplexOrder BigOperators Matrix.Norms.L2Operator |
| open Matrix |
| open Filter Topology |
| open PolynomialPEPS.PinnedEntropy |
|
|
| open Set Filter Complex Complex.HadamardThreeLines |
| open scoped Topology |
|
|
| namespace PolynomialPEPS.Subvolume.FilterInterpolation |
|
|
| theorem deriv_le_of_right_comparison {f g : ℝ → ℝ} {f' g' : ℝ} |
| (hf : HasDerivAt f f' 0) (hg : HasDerivAt g g' 0) |
| (hzero : f 0 = g 0) (hfg : ∀ x ∈ Icc (0 : ℝ) 1, f x ≤ g x) : |
| f' ≤ g' := by |
| have hfl := (hasDerivAt_iff_tendsto_slope_left_right.mp hf).2 |
| have hgl := (hasDerivAt_iff_tendsto_slope_left_right.mp hg).2 |
| apply le_of_tendsto_of_tendsto hfl hgl |
| filter_upwards [self_mem_nhdsWithin, |
| (eventually_lt_nhds (show (0 : ℝ) < 1 by norm_num)).filter_mono nhdsWithin_le_nhds] |
| with x hx hx1 |
| simp only [mem_Ioi] at hx |
| simp only [slope_def_field, sub_zero] |
| apply div_le_div_of_nonneg_right _ (le_of_lt hx) |
| rw [hzero] |
| exact sub_le_sub_right (hfg x ⟨le_of_lt hx, le_of_lt hx1⟩) (g 0) |
|
|
| theorem threeLines_tangent_zero {f : ℂ → ℂ} {d : ℂ} {N : ℝ} |
| (hN : 0 < N) (hzero : f 0 = 1) (hderiv : HasDerivAt f d 0) |
| (hd : DiffContOnCl ℂ f (verticalStrip 0 1)) |
| (hB : BddAbove ((norm ∘ f) '' verticalClosedStrip 0 1)) |
| (hleft : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ 1) |
| (hright : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ N) : |
| d.re ≤ Real.log N := by |
| have hf : HasDerivAt (fun x : ℝ => (f x).re) d.re 0 := by |
| simpa using hderiv.real_of_complex |
| have hg : HasDerivAt (fun x : ℝ => Real.exp (x * Real.log N)) (Real.log N) 0 := by |
| convert ((hasDerivAt_id (0 : ℝ)).mul_const (Real.log N)).exp using 1 <;> simp |
| apply deriv_le_of_right_comparison hf hg |
| · simp [hzero] |
| · intro x hx |
| have hz : (x : ℂ) ∈ verticalClosedStrip 0 1 := by simpa [verticalClosedStrip] using hx |
| have hbound := HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip₀₁' |
| f hz hd hB hleft hright |
| have habs : (f x).re ≤ ‖f x‖ := Complex.re_le_norm _ |
| refine habs.trans (hbound.trans_eq ?_) |
| simp only [Complex.ofReal_re, Real.one_rpow, one_mul] |
| rw [Real.rpow_def_of_pos hN] |
| congr 1 |
| ring |
| |
| theorem entropy_cost_upper_of_spectral_curve {f : ℂ → ℂ} {S N : ℝ} |
| (hN : 0 < N) (hzero : f 0 = 1) |
| (hderiv : HasDerivAt f ((-S / 2 : ℝ) : ℂ) 0) |
| (hd : DiffContOnCl ℂ f (verticalStrip 0 1)) |
| (hB : BddAbove ((norm ∘ f) '' verticalClosedStrip 0 1)) |
| (hleft : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ 1) |
| (hright : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ N) : |
| -Real.log (N ^ 2) ≤ S := by |
| have h := threeLines_tangent_zero hN hzero hderiv hd hB hleft hright |
| simp only [Complex.ofReal_re] at h |
| rw [Real.log_pow] |
| norm_num only [Nat.cast_ofNat] |
| linarith |
| |
| end PolynomialPEPS.Subvolume.FilterInterpolation |
| |
| end |
| |
| end OAI |
| |
