openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /GraphSourceFixedGeometry.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.TwoRpowNMul | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFixedSpatialTerm | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| section | |
| section | |
| namespace Attempt4FullFar | |
| open MeasureTheory Set Metric Filter Real GraphSourceScale | |
| open scoped ContDiff RealInnerProductSpace | |
| lemma graph_source_fixed_geometry (N a p v sig : ℝ) (G c₀ : ℝ) | |
| (_hc₀ : 0 < c₀) : | |
| let R := (2:ℝ)^N | |
| let H := (2:ℝ)^(p-a) | |
| let V := (2:ℝ)^(v-p) | |
| let δ := min c₀ (H⁻¹*R^(10*sig)) | |
| let τ := (2:ℝ)^(-p)*R^(30*sig) | |
| let D := G*(2:ℝ)^(-a) | |
| ((2:ℝ)^v*τ=V*R^(30*sig)) ∧ | |
| (D/τ=G*H*R^(-30*sig)) ∧ | |
| (δ=min c₀ ((2:ℝ)^a/(2:ℝ)^p*R^(10*sig))) := by | |
| dsimp only | |
| have ht : 0 < (2:ℝ)^N := by positivity | |
| constructor | |
| · rw [←mul_assoc,←Real.rpow_add (by norm_num)]; congr 2 | |
| constructor | |
| · rw [div_eq_mul_inv,mul_inv_rev,←Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),neg_neg, | |
| ←Real.rpow_neg ht.le] | |
| rw [show G*(2:ℝ)^(-a)*(((2:ℝ)^N)^(-(30*sig))*(2:ℝ)^p) = | |
| G*((2:ℝ)^p*(2:ℝ)^(-a))*((2:ℝ)^N)^(-30*sig) by ring_nf] | |
| rw [←Real.rpow_add (by norm_num)]; congr 2 | |
| · rw [←Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),neg_sub, | |
| Real.rpow_sub (by norm_num)] | |
| theorem graph_source_fixed_nonedge_decay | |
| (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| {sig c₀ Ccap G : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀) | |
| (hCcap : 0 ≤ Ccap) (hG : 0 ≤ G) (T : ℝ) : | |
| ∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, | |
| ∀ (a p v L : ℝ), p-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p → | |
| let R := (2:ℝ)^N | |
| let δ := min c₀ (((2:ℝ)^(p-a))⁻¹*R^(10*sig)) | |
| let τ := (2:ℝ)^(-p)*R^(30*sig) | |
| let A := 1+R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))+δ⁻¹ | |
| ∀ (Ψ : ℝ → ℂ) (w₁ w₂ d₁ d₂ e₁ e₂ : V) (K B₁ B₂ : ℝ) (b₁ b₂ : V → ℂ), | |
| ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 → | |
| (∫ u : ℝ, ‖Ψ u‖) ≤ K → (∀ u, ‖iteratedDeriv m Ψ u‖ ≤ K) → | |
| ‖w₁‖=1 → ‖w₂‖=1 → 0 ≤ K → 0 ≤ B₁ → 0 ≤ B₂ → | |
| ‖e₁‖ ≤ G*(2:ℝ)^(-a) → ‖e₂‖ ≤ G*(2:ℝ)^(-a) → | |
| ‖e₁-d₁‖ ≤ G*(2:ℝ)^(-p) → ‖e₂-d₂‖ ≤ G*(2:ℝ)^(-p) → | |
| ¬(‖d₁-inner ℝ d₁ w₁ • w₁‖ ≤ τ ∧ | |
| ‖d₂-inner ℝ d₂ w₂ • w₂‖ ≤ τ ∧ |inner ℝ w₁ d₁+inner ℝ w₂ d₂| ≤ τ) → | |
| ContDiff ℝ ∞ b₁ → ContDiff ℝ ∞ b₂ → | |
| (∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j b₁ w‖ ≤ B₁*A^j) → | |
| (∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j b₂ w‖ ≤ B₂*A^j) → | |
| (∀ w, ‖w‖=1 → b₁ w ≠ 0 → ‖w-w₁‖ ≤ Ccap*δ) → | |
| (∀ w, ‖w‖=1 → b₂ w ≠ 0 → ‖w-w₂‖ ≤ Ccap*δ) → | |
| ‖graphFixedSpatialTermND Ψ ((2:ℝ)^v) e₁ e₂ b₁ b₂‖ ≤ C*K*B₁*B₂*R^(-T) := by | |
| obtain ⟨m,hm⟩ := exists_nat_gt (T/(29*sig)) | |
| have hm' : T ≤ 29*sig*m := by | |
| have hh := (div_lt_iff₀ (by positivity : 0 < 29*sig)).mp hm | |
| linarith | |
| obtain ⟨B,hB,hBb⟩ := graph_fixed_nonedge_term_uniform V m | |
| have hbase : 0 ≤ 5+c₀⁻¹+G := by positivity | |
| refine ⟨m,B*(5+c₀⁻¹+G)^m,by positivity,?_⟩ | |
| have hpow : Tendsto (fun N : ℕ => (2:ℝ)^N) atTop atTop := | |
| tendsto_pow_atTop_atTop_of_one_lt (by norm_num) | |
| obtain ⟨N₀,hN₀⟩ := Filter.eventually_atTop.mp | |
| (((tendsto_rpow_atTop (by positivity : 0 < 20*sig)).comp hpow).eventually | |
| (Filter.eventually_ge_atTop (8*Ccap*G+4*G))) | |
| refine ⟨N₀,?_⟩ | |
| intro N hN a p v L hh hL hq R δ τ A Ψ w₁ w₂ d₁ d₂ e₁ e₂ K B₁ B₂ b₁ b₂ | |
| hΨ hsΨ hK hdΨ hw₁ hw₂ hK0 hB₁ hB₂ he₁ he₂ hed₁ hed₂ hne hb₁ hb₂ hd₁ hd₂ hs₁ hs₂ | |
| have hR : 1 ≤ R := one_le_pow₀ (by norm_num) | |
| have hRp : 0 < R := lt_of_lt_of_le zero_lt_one hR | |
| have hδ : 0 < δ := lt_min hc₀ (by positivity) | |
| have hτ : 0 < τ := by dsimp [τ]; positivity | |
| have hA : 1 ≤ A := by | |
| have : 0 ≤ R^sig*max 1 ((2:ℝ)^L*R^(-50*sig)) := by positivity | |
| dsimp [A]; linarith [inv_pos.mpr hδ] | |
| have hD : 0 ≤ G*(2:ℝ)^(-a) := by positivity | |
| have hη : 0 ≤ G*(2:ℝ)^(-p) := by positivity | |
| have hg := graph_source_fixed_geometry (N:ℝ) a p v sig G c₀ hc₀ | |
| simp only [Real.rpow_natCast] at hg | |
| rcases hg with ⟨hQτ,hDτ,hδeq⟩ | |
| have herr : 4*(Ccap*δ)*(G*(2:ℝ)^(-a))+2*(G*(2:ℝ)^(-p)) ≤ τ/2 := by | |
| dsimp only [δ,τ,R] | |
| rw [hδeq] | |
| simp only [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),←div_eq_mul_inv] | |
| exact source_error_scale hR hsig.le (by positivity) (by positivity) hCcap hG (hN₀ N hN) | |
| have hterm := hBb Ψ w₁ w₂ d₁ d₂ e₁ e₂ ((2:ℝ)^v) τ (Ccap*δ) | |
| (G*(2:ℝ)^(-a)) (G*(2:ℝ)^(-p)) A K B₁ B₂ b₁ b₂ | |
| hΨ hsΨ hK hdΨ hw₁ hw₂ (by positivity) hτ (by positivity) hD hη hA | |
| hK0 hB₁ hB₂ he₁ he₂ hed₁ hed₂ herr hne hb₁ hb₂ hd₁ hd₂ hs₁ hs₂ | |
| obtain ⟨hH,hU⟩ := length_scale hh hL | |
| simp only [Real.rpow_natCast] at hH hU | |
| have hV : 1 ≤ (2:ℝ)^(v-p) := Real.one_le_rpow (by norm_num) hq | |
| have hratio := ratio_power_decay hc₀ hR hsig.le (by positivity : 0 < (2:ℝ)^(p-a)) hV | |
| hG (by positivity : 0 ≤ (2:ℝ)^L) hH hU hm' | |
| have heq : (A+1+G*(2:ℝ)^(-a)/τ)/((2:ℝ)^v*τ) = | |
| (2+R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))+δ⁻¹+ | |
| G*(2:ℝ)^(p-a)*R^(-30*sig))/((2:ℝ)^(v-p)*R^(30*sig)) := by | |
| rw [hQτ,hDτ]; dsimp [A]; ring | |
| rw [heq] at hterm | |
| apply hterm.trans | |
| calc | |
| _ ≤ B*K*B₁*B₂*((5+c₀⁻¹+G)^m*R^(-T)) := mul_le_mul_of_nonneg_left hratio (by positivity) | |
| _ = _ := by ring | |
| end Attempt4FullFar | |
| end | |
| end | |
| end OAI | |