openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /GraphFiniteHFBound2.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Angular.AngularHFSmul2 | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFiniteCutoffEstimate2 | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFiniteHFBound | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4Near | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate | |
| variable {V α β κ : Type*} [MeasurableSpace V] | |
| theorem graphFiniteHFND_bound (d : ℕ) (a p v lam₁ lam₂ : ℝ) | |
| (μ : Measure V) [IsFiniteMeasure μ] | |
| (A : Finset α) (B : Finset β) (K : Finset κ) | |
| (G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)] | |
| (η : κ → V × V → ℝ) | |
| (hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1) | |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) | |
| (hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j)) | |
| (C : α → ℝ) (E : β → ℝ) | |
| (hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i) | |
| (hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) {D : ℝ} (hD : 0 ≤ D) | |
| (hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D) | |
| (hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D) | |
| {Ψ : ℝ → ℝ} (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) (hΨpos : ∀ r, 0 ≤ Ψ r) | |
| {ψ : V → ℂ} (hmψ : Measurable ψ) (hψ : ∀ᵐ w ∂μ, ‖ψ w‖ ≤ 1) | |
| (hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, | |
| Integrable (fun p : ℝ × (V × V) => (Ψ (p.1/(2:ℝ)^v):ℂ)*(η k p.2:ℂ)* | |
| (((ψ p.2.1*U a.1 (p.1,p.2.1))*W a.2 (p.1,p.2.2))* | |
| conj ((ψ p.2.2*U b.1 (p.1,p.2.2))*W b.2 (p.1,p.2.1)))) (volume.prod (μ.prod μ))) | |
| {ε : ℝ} (hε : 0 ≤ ε) | |
| (hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, ¬G k a b → | |
| ‖∫ p : ℝ × (V × V), (Ψ (p.1/(2:ℝ)^v):ℂ)*(η k p.2:ℂ)* | |
| (((ψ p.2.1*U a.1 (p.1,p.2.1))*W a.2 (p.1,p.2.2))* | |
| conj ((ψ p.2.2*U b.1 (p.1,p.2.2))*W b.2 (p.1,p.2.1))) ∂volume.prod (μ.prod μ)‖ ≤ ε) : | |
| sourceAngularHFND d μ a v Ψ ψ (fun z => lam₁ • ∑ i ∈ A, U i z) | |
| (fun z => lam₂ • ∑ j ∈ B, W j z) ≤ | |
| lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D* | |
| (∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ p v Ψ (U i) (W j)) + | |
| lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹* | |
| ((K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε) := by | |
| have h := graphFiniteCutoffEstimateND μ A B K G η hpart U W hU hW C E hbU hbW | |
| hD hrow hcol hΨ (fun r => hΨpos (r/(2:ℝ)^v)) hmψ hψ hi hε hne | |
| have he : (2:ℝ)^((d:ℝ)*(p-a))*(∑ i ∈ A, ∑ j ∈ B, | |
| sourceAngularHND d μ p v Ψ (U i) (W j)) = | |
| ∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ a v Ψ (U i) (W j) := by | |
| simp_rw [Finset.mul_sum,← sourceAngularHND_depth d μ a p v Ψ] | |
| have hm := mul_le_mul_of_nonneg_left h | |
| (show 0 ≤ lam₁^2*lam₂^2*((2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹) by positivity) | |
| rw [sourceAngularHFND_smul] | |
| have hr : lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D* | |
| (∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ p v Ψ (U i) (W j)) = | |
| lam₁^2*lam₂^2*D*(∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ a v Ψ (U i) (W j)) := by | |
| rw [← he] | |
| ring | |
| rw [hr] | |
| simp only [sourceAngularHND,sourceAngularHFND,← Finset.mul_sum] at * | |
| nlinarith only [hm] | |
| end Attempt4Near | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4Near | |
| open Set MeasureTheory | |
| variable {V α β : Type*} [MeasurableSpace V] | |
| noncomputable def graphChildMaxND (d : ℕ) (μ : Measure V) (p v : ℝ) | |
| (Ψ : ℝ → ℝ) (A : Finset α) (B : Finset β) (hAB : (A.product B).Nonempty) | |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) : ℝ := | |
| (A.product B).sup' hAB (fun q => sourceAngularHND d μ p v Ψ (U q.1) (W q.2)) | |
| lemma graphChildMaxND_nonneg (d : ℕ) (μ : Measure V) (p v : ℝ) | |
| (Ψ : ℝ → ℝ) (hΨ : ∀ r, 0 ≤ Ψ r) | |
| (A : Finset α) (B : Finset β) (hAB : (A.product B).Nonempty) | |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) : | |
| 0 ≤ graphChildMaxND d μ p v Ψ A B hAB U W := by | |
| obtain ⟨q,hq⟩ := hAB | |
| exact (sourceAngularHND_nonneg d μ p v hΨ (U q.1) (W q.2)).trans | |
| (Finset.le_sup' (fun q => sourceAngularHND d μ p v Ψ (U q.1) (W q.2)) hq) | |
| lemma graphChild_sum_le_maxND (d : ℕ) (μ : Measure V) (p v : ℝ) | |
| (Ψ : ℝ → ℝ) (A : Finset α) (B : Finset β) (hAB : (A.product B).Nonempty) | |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) : | |
| (∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ p v Ψ (U i) (W j)) ≤ | |
| ((A.card:ℝ)*B.card)*graphChildMaxND d μ p v Ψ A B hAB U W := by | |
| rw [← Finset.sum_product (f := fun q : α × β => sourceAngularHND d μ p v Ψ (U q.1) (W q.2))] | |
| calc | |
| _ ≤ ∑ _q ∈ A.product B, graphChildMaxND d μ p v Ψ A B hAB U W := | |
| Finset.sum_le_sum (fun q hq => Finset.le_sup' | |
| (fun q => sourceAngularHND d μ p v Ψ (U q.1) (W q.2)) hq) | |
| _ = _ := by simp [Finset.card_product,mul_assoc] | |
| end Attempt4Near | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4Near | |
| lemma graph_profile_prefactorND {N : ℕ} (hN : 0 < N) (d a p : ℝ) : | |
| (2:ℝ)^(d*(p-a)) = ((2:ℝ)^N)^(d*(p-a)/(N:ℝ)) := by | |
| rw [← Real.rpow_natCast (2:ℝ) N,← Real.rpow_mul (by norm_num : (0:ℝ) ≤ 2)] | |
| congr 1 | |
| have hn : (N:ℝ) ≠ 0 := by positivity | |
| field_simp | |
| lemma graph_profile_cancellationND {N : ℕ} (hN : 0 < N) (d a p Δ₁ Δ₂ loss : ℝ) : | |
| let R : ℝ := (2:ℝ)^N | |
| let I₁ := Δ₁+(d/2)*(p-a)/(N:ℝ) | |
| let I₂ := Δ₂+(d/2)*(p-a)/(N:ℝ) | |
| (R^(-I₁))^2*(R^(-I₂))^2*R^(I₁+I₂+loss)*(2:ℝ)^(d*(p-a)) = | |
| R^(-(Δ₁+Δ₂)+loss) := by | |
| dsimp only | |
| have hR : 0 < (2:ℝ)^N := by positivity | |
| rw [graph_profile_prefactorND hN] | |
| simp only [← Real.rpow_mul_natCast hR.le,← Real.rpow_add hR] | |
| congr 1 | |
| push_cast | |
| ring | |
| theorem graph_profile_coefficient_leND {N : ℕ} (hN : 0 < N) | |
| (d a p Δ₁ Δ₂ loss degree : ℝ) {D L C : ℝ} | |
| (hD : 0 ≤ D) (hDL : D ≤ C*((2:ℝ)^N)^degree) | |
| (hL : L ≤ ((2:ℝ)^N)^(Δ₁+Δ₂+d*(p-a)/(N:ℝ)+loss)) : | |
| ((((2:ℝ)^N)^(-(Δ₁+(d/2)*(p-a)/(N:ℝ))))^2* | |
| (((2:ℝ)^N)^(-(Δ₂+(d/2)*(p-a)/(N:ℝ))))^2)* | |
| (2:ℝ)^(d*(p-a))*D*L ≤ | |
| C*((2:ℝ)^N)^(-(Δ₁+Δ₂)+degree+loss) := by | |
| let R : ℝ := (2:ℝ)^N | |
| have hR : 0 < R := by positivity | |
| let I₁ := Δ₁+(d/2)*(p-a)/(N:ℝ) | |
| let I₂ := Δ₂+(d/2)*(p-a)/(N:ℝ) | |
| have he : Δ₁+Δ₂+d*(p-a)/(N:ℝ)=I₁+I₂ := by dsimp [I₁,I₂]; ring | |
| have h := mul_le_mul_of_nonneg_left | |
| (mul_le_mul_of_nonneg_left hL hD) (show 0 ≤ (R^(-I₁))^2*(R^(-I₂))^2*(2:ℝ)^(d*(p-a)) by positivity) | |
| have h' := mul_le_mul_of_nonneg_right hDL (show 0 ≤ R^(I₁+I₂+loss) by positivity) | |
| have h'' := mul_le_mul_of_nonneg_left h' | |
| (show 0 ≤ (R^(-I₁))^2*(R^(-I₂))^2*(2:ℝ)^(d*(p-a)) by positivity) | |
| change (R^(-I₁))^2*(R^(-I₂))^2*(2:ℝ)^(d*(p-a))*D*L ≤ _ | |
| change L ≤ R^(Δ₁+Δ₂+d*(p-a)/(N:ℝ)+loss) at hL | |
| rw [he] at hL | |
| calc | |
| _ ≤ (R^(-I₁))^2*(R^(-I₂))^2*(2:ℝ)^(d*(p-a))*(D*R^(I₁+I₂+loss)) := by | |
| simpa only [mul_assoc,he] using h | |
| _ ≤ (R^(-I₁))^2*(R^(-I₂))^2*(2:ℝ)^(d*(p-a))*((C*R^degree)*R^(I₁+I₂+loss)) := h'' | |
| _ = C*((R^(-I₁))^2*(R^(-I₂))^2*R^(I₁+I₂+loss)*(2:ℝ)^(d*(p-a)))*R^degree := by ring | |
| _ = C*R^(-(Δ₁+Δ₂)+loss)*R^degree := by rw [graph_profile_cancellationND hN] | |
| _ = _ := by rw [mul_assoc,← Real.rpow_add hR]; congr 2; ring | |
| end Attempt4Near | |
| end GraphLiteralSupport | |
| end | |
| end | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4Near | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate | |
| variable {V α β κ : Type*} [MeasurableSpace V] | |
| theorem graphFiniteHND_to_child_max (d : ℕ) (a p v lam₁ lam₂ : ℝ) {N : ℕ} (hN : 0<N) | |
| (Δ₁ Δ₂ degree loss C₀ : ℝ) | |
| (hlam₁ : lam₁=((2:ℝ)^N)^(-(Δ₁+(d:ℝ)/2*(p-a)/(N:ℝ)))) | |
| (hlam₂ : lam₂=((2:ℝ)^N)^(-(Δ₂+(d:ℝ)/2*(p-a)/(N:ℝ)))) | |
| (μ : Measure V) [IsFiniteMeasure μ] | |
| (A : Finset α) (B : Finset β) (K : Finset κ) (hAB : (A.product B).Nonempty) | |
| (hlabels : ((A.card:ℝ)*B.card) ≤ ((2:ℝ)^N)^(Δ₁+Δ₂+(d:ℝ)*(p-a)/(N:ℝ)+loss)) | |
| (G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)] | |
| (η : κ → V × V → ℝ) | |
| (hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1) | |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) | |
| (hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j)) | |
| (C : α → ℝ) (E : β → ℝ) | |
| (hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i) | |
| (hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) (D : ℝ) (hD : 0 ≤ D) | |
| (hdegree : D ≤ C₀*((2:ℝ)^N)^degree) | |
| (hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D) | |
| (hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D) | |
| {Ψ : ℝ → ℝ} (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) (hΨpos : ∀ r, 0 ≤ Ψ r) | |
| (hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, | |
| Integrable (fun p : ℝ × (V × V) => (Ψ (p.1/(2:ℝ)^v):ℂ)*(η k p.2:ℂ)* | |
| ((U a.1 (p.1,p.2.1)*W a.2 (p.1,p.2.2))* | |
| conj (U b.1 (p.1,p.2.1)*W b.2 (p.1,p.2.2)))) (volume.prod (μ.prod μ))) | |
| {ε : ℝ} (hε : 0 ≤ ε) | |
| (hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, ¬G k a b → | |
| ‖∫ p : ℝ × (V × V), (Ψ (p.1/(2:ℝ)^v):ℂ)*(η k p.2:ℂ)* | |
| ((U a.1 (p.1,p.2.1)*W a.2 (p.1,p.2.2))* | |
| conj (U b.1 (p.1,p.2.1)*W b.2 (p.1,p.2.2))) ∂volume.prod (μ.prod μ)‖ ≤ ε) : | |
| sourceAngularHND d μ a v Ψ (fun z => lam₁ • ∑ i ∈ A, U i z) | |
| (fun z => lam₂ • ∑ j ∈ B, W j z) ≤ | |
| C₀*((2:ℝ)^N)^(-(Δ₁+Δ₂)+degree+loss)* | |
| graphChildMaxND d μ p v Ψ A B hAB U W + | |
| lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹* | |
| ((K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε) := by | |
| have h := graphFiniteHND_bound d a p v lam₁ lam₂ μ A B K G η hpart U W hU hW C E hbU hbW | |
| D hrow hcol hΨ hΨpos hi hε hne | |
| have hs := graphChild_sum_le_maxND d μ p v Ψ A B hAB U W | |
| have hmax := graphChildMaxND_nonneg d μ p v Ψ hΨpos A B hAB U W | |
| have hc : lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D*((A.card:ℝ)*B.card) ≤ | |
| C₀*((2:ℝ)^N)^(-(Δ₁+Δ₂)+degree+loss) := by | |
| rw [hlam₁,hlam₂] | |
| exact graph_profile_coefficient_leND hN d a p Δ₁ Δ₂ loss degree hD hdegree hlabels | |
| apply h.trans | |
| apply add_le_add _ le_rfl | |
| calc | |
| _ ≤ (lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D)* | |
| (((A.card:ℝ)*B.card)*graphChildMaxND d μ p v Ψ A B hAB U W) := | |
| mul_le_mul_of_nonneg_left hs (by positivity) | |
| _ = (lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D*((A.card:ℝ)*B.card))* | |
| graphChildMaxND d μ p v Ψ A B hAB U W := by ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_right hc hmax | |
| theorem graphFiniteHFND_to_child_max (d : ℕ) (a p v lam₁ lam₂ : ℝ) {N : ℕ} (hN : 0<N) | |
| (Δ₁ Δ₂ degree loss C₀ : ℝ) | |
| (hlam₁ : lam₁=((2:ℝ)^N)^(-(Δ₁+(d:ℝ)/2*(p-a)/(N:ℝ)))) | |
| (hlam₂ : lam₂=((2:ℝ)^N)^(-(Δ₂+(d:ℝ)/2*(p-a)/(N:ℝ)))) | |
| (μ : Measure V) [IsFiniteMeasure μ] | |
| (A : Finset α) (B : Finset β) (K : Finset κ) (hAB : (A.product B).Nonempty) | |
| (hlabels : ((A.card:ℝ)*B.card) ≤ ((2:ℝ)^N)^(Δ₁+Δ₂+(d:ℝ)*(p-a)/(N:ℝ)+loss)) | |
| (G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)] | |
| (η : κ → V × V → ℝ) | |
| (hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1) | |
| (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) | |
| (hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j)) | |
| (C : α → ℝ) (E : β → ℝ) | |
| (hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i) | |
| (hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) {D : ℝ} (hD : 0 ≤ D) | |
| (hdegree : D ≤ C₀*((2:ℝ)^N)^degree) | |
| (hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D) | |
| (hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D) | |
| {Ψ : ℝ → ℝ} (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) (hΨpos : ∀ r, 0 ≤ Ψ r) | |
| {ψ : V → ℂ} (hmψ : Measurable ψ) (hψ : ∀ᵐ w ∂μ, ‖ψ w‖ ≤ 1) | |
| (hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, | |
| Integrable (fun p : ℝ × (V × V) => (Ψ (p.1/(2:ℝ)^v):ℂ)*(η k p.2:ℂ)* | |
| (((ψ p.2.1*U a.1 (p.1,p.2.1))*W a.2 (p.1,p.2.2))* | |
| conj ((ψ p.2.2*U b.1 (p.1,p.2.2))*W b.2 (p.1,p.2.1)))) (volume.prod (μ.prod μ))) | |
| {ε : ℝ} (hε : 0 ≤ ε) | |
| (hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, ¬G k a b → | |
| ‖∫ p : ℝ × (V × V), (Ψ (p.1/(2:ℝ)^v):ℂ)*(η k p.2:ℂ)* | |
| (((ψ p.2.1*U a.1 (p.1,p.2.1))*W a.2 (p.1,p.2.2))* | |
| conj ((ψ p.2.2*U b.1 (p.1,p.2.2))*W b.2 (p.1,p.2.1))) ∂volume.prod (μ.prod μ)‖ ≤ ε) : | |
| sourceAngularHFND d μ a v Ψ ψ (fun z => lam₁ • ∑ i ∈ A, U i z) | |
| (fun z => lam₂ • ∑ j ∈ B, W j z) ≤ | |
| C₀*((2:ℝ)^N)^(-(Δ₁+Δ₂)+degree+loss)* | |
| graphChildMaxND d μ p v Ψ A B hAB U W + | |
| lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹* | |
| ((K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε) := by | |
| have h := graphFiniteHFND_bound d a p v lam₁ lam₂ μ A B K G η hpart U W hU hW C E hbU hbW | |
| hD hrow hcol hΨ hΨpos hmψ hψ hi hε hne | |
| have hs := graphChild_sum_le_maxND d μ p v Ψ A B hAB U W | |
| have hmax := graphChildMaxND_nonneg d μ p v Ψ hΨpos A B hAB U W | |
| have hc : lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D*((A.card:ℝ)*B.card) ≤ | |
| C₀*((2:ℝ)^N)^(-(Δ₁+Δ₂)+degree+loss) := by | |
| rw [hlam₁,hlam₂] | |
| exact graph_profile_coefficient_leND hN d a p Δ₁ Δ₂ loss degree hD hdegree hlabels | |
| apply h.trans | |
| apply add_le_add _ le_rfl | |
| calc | |
| _ ≤ (lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D)* | |
| (((A.card:ℝ)*B.card)*graphChildMaxND d μ p v Ψ A B hAB U W) := | |
| mul_le_mul_of_nonneg_left hs (by positivity) | |
| _ = (lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D*((A.card:ℝ)*B.card))* | |
| graphChildMaxND d μ p v Ψ A B hAB U W := by ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_right hc hmax | |
| end Attempt4Near | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |