| import OAI.Geometry.RCD.Harmonic.EventualQuadraticError |
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| namespace OAI |
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| /-! # Radius-preserving balance of the quadratic Young parameter -/ |
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|
| open Filter |
| open scoped Topology |
| noncomputable section |
| namespace RegularPointCoordinates |
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|
| theorem eventual_balanced_error {Y : Type*} (f : ℕ → Y → ℝ) |
| {C V M B : ℝ} (hV : 0<V) (hM : 0<M) (hB : 0≤B) |
| (h : ∀ δ : ℝ, 0<δ → ∀ᶠ k in atTop, ∀ y (a : ℝ), 0<a → |
| 4*a*|f k y|≤C*(4*a^2*V+M*(B+δ))) {δ : ℝ} (hδ : 0<δ) : |
| ∀ᶠ k in atTop, ∀ y, |f k y|≤(C*Real.sqrt V*Real.sqrt M)*Real.sqrt (B+δ) := by |
| let v := Real.sqrt V |
| let m := Real.sqrt M |
| let b := Real.sqrt (B+δ) |
| have hv : 0<v := Real.sqrt_pos.mpr hV |
| have hm : 0<m := Real.sqrt_pos.mpr hM |
| have hb : 0<b := Real.sqrt_pos.mpr (by linarith) |
| have hv2 : v^2=V := Real.sq_sqrt hV.le |
| have hm2 : m^2=M := Real.sq_sqrt hM.le |
| have hb2 : b^2=B+δ := Real.sq_sqrt (by linarith) |
| let a := m*b/(2*v) |
| have ha : 0<a := div_pos (mul_pos hm hb) (by positivity) |
| have he : C*(4*a^2*V+M*(B+δ))=(4*a)*((C*v*m)*b) := by |
| calc |
| _ = C*(4*(m*b/(2*v))^2*v^2+m^2*b^2) := by rw [hv2,hm2,hb2] |
| _ = _ := by |
| dsimp only [a] |
| field_simp [hv.ne'] |
| ring |
| filter_upwards [h δ hδ] with k hk y |
| have hh := hk y a ha |
| rw [he] at hh |
| exact (mul_le_mul_iff_right₀ (show 0<4*a by positivity)).mp hh |
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|
| theorem eventual_balanced_error_add {Y : Type*} (f : ℕ → Y → ℝ) |
| {C V M B : ℝ} (hV : 0<V) (hM : 0<M) (hB : 0≤B) |
| (h : ∀ δ : ℝ, 0<δ → ∀ᶠ k in atTop, ∀ y (a : ℝ), 0<a → |
| 4*a*|f k y|≤C*(4*a^2*V+M*(B+δ))) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ k in atTop, ∀ y, |
| |f k y|≤(C*Real.sqrt V*Real.sqrt M)*Real.sqrt B+ε := by |
| let δ (j : ℕ) : ℝ := 1/((j:ℝ)+1) |
| have hδ j : 0<δ j := by dsimp only [δ]; positivity |
| have hz : Tendsto δ atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat |
| have hs := ((tendsto_const_nhds (x := B)).add hz).sqrt.const_mul |
| (C*Real.sqrt V*Real.sqrt M) |
| simp only [add_zero] at hs |
| obtain ⟨j,hj⟩ := (hs.eventually (gt_mem_nhds (show |
| (C*Real.sqrt V*Real.sqrt M)*Real.sqrt B< |
| (C*Real.sqrt V*Real.sqrt M)*Real.sqrt B+ε by linarith))).exists |
| filter_upwards [eventual_balanced_error f hV hM hB h (hδ j)] with k hk y |
| exact (hk y).trans hj.le |
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|
| end RegularPointCoordinates |
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| end |
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| end OAI |
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