| import OAI.Geometry.PinnedDistances.ArithmeticGraphs |
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| namespace OAI |
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| universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 |
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| section |
| noncomputable section |
| open NumberField |
| open scoped BigOperators |
| noncomputable section |
| open MeasureTheory |
| open scoped Topology |
| noncomputable section |
| open MeasureTheory |
| open scoped Topology ENNReal |
| noncomputable section |
| open MeasureTheory NumberField |
| open scoped BigOperators |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory |
| noncomputable section |
| open Filter |
| open scoped Topology |
| noncomputable section |
| open scoped BigOperators |
| noncomputable section |
| open Filter |
| open scoped Topology |
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| namespace WeakPinned |
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| /-- The scalar limiting step for small overlap, with fixed positive τ and H. -/ |
|
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| theorem small_overlap_rate (n η U A D : ℕ → ℝ) (θ : ℝ) (hθ : 0 < θ) |
| (hn : Tendsto n atTop atTop) (hη : Tendsto η atTop (𝓝 0)) |
| (hη0 : ∀ j, 0 ≤ η j) (hU : ∀ j, 0 ≤ U j) (hA : ∀ j, 0 ≤ A j) (hD : ∀ j, 0 ≤ D j) |
| (hbound : ∀ τ H : ℝ, 0 < τ → τ ≤ 1 → 0 ≤ H → ∀ᶠ j in atTop, |
| D j ≤ (2*(1/Real.log (n j))/τ^2+2*(1/Real.log (n j))/(θ-η j)+((θ+η j)/(θ-η j)-1))*U j + |
| (2*τ/(θ-η j)+(2-2*((θ-η j)/(θ+η j/8))))*A j + |
| 4*(1/Real.log (n j))*H + 32*Real.exp (-H/2)) : |
| Tendsto (fun j => D j/(U j+A j+1)) atTop (𝓝 0) := by |
| have hlamT : Tendsto (fun j => 1/Real.log (n j)) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop (Real.tendsto_log_atTop.comp hn) |
| apply tendsto_order.mpr |
| constructor |
| · intro a ha |
| exact Filter.Eventually.of_forall fun j => ha.trans_le (div_nonneg (hD j) (by linarith [hU j,hA j])) |
| · intro ε hε |
| let τ := min 1 (ε*θ/16) |
| have hτ : 0 < τ := lt_min (by norm_num) (by positivity) |
| have hτ1 : τ ≤ 1 := min_le_left _ _ |
| have hτb : 2*τ/θ ≤ ε/8 := by |
| apply (div_le_iff₀ hθ).mpr |
| have := min_le_right (1:ℝ) (ε*θ/16) |
| dsimp [τ] at ⊢ |
| linarith |
| have htExp : Tendsto (fun H : ℝ => 32*Real.exp (-H/2)) atTop (𝓝 0) := by |
| have hd : Tendsto (fun H : ℝ => H/2) atTop atTop := by |
| simpa only [div_eq_mul_inv, mul_comm, id_eq, one_mul] using |
| (tendsto_const_mul_atTop_of_pos (by norm_num : (0:ℝ)<1/2)).mpr tendsto_id |
| simpa only [Function.comp_def,neg_div,mul_zero] using |
| (Real.tendsto_exp_neg_atTop_nhds_zero.comp hd).const_mul 32 |
| obtain ⟨H,hH,hHb⟩ := (eventually_ge_atTop (0:ℝ) |>.and (htExp.eventually (eventually_lt_nhds (show (0:ℝ)<ε/8 by linarith)))).exists |
| let a := fun j => 2*(1/Real.log (n j))/τ^2+2*(1/Real.log (n j))/(θ-η j)+((θ+η j)/(θ-η j)-1) |
| let b := fun j => 2*τ/(θ-η j)+(2-2*((θ-η j)/(θ+η j/8))) |
| let e := fun j => 4*(1/Real.log (n j))*H+32*Real.exp (-H/2) |
| have hθm : Tendsto (fun j => θ-η j) atTop (𝓝 θ) := by |
| simpa using (tendsto_const_nhds (x:=θ)).sub hη |
| have hθp : Tendsto (fun j => θ+η j) atTop (𝓝 θ) := by |
| simpa using (tendsto_const_nhds (x:=θ)).add hη |
| have hθp8 : Tendsto (fun j => θ+η j/8) atTop (𝓝 θ) := by |
| simpa using (tendsto_const_nhds (x:=θ)).add (hη.div_const 8) |
| have ha : Tendsto a atTop (𝓝 0) := by |
| have ha1 : Tendsto (fun j => 2*(1/Real.log (n j))/τ^2) atTop (𝓝 0) := by |
| simpa using (hlamT.const_mul 2).div_const (τ^2) |
| have ha2 : Tendsto (fun j => 2*(1/Real.log (n j))/(θ-η j)) atTop (𝓝 0) := by |
| convert (hlamT.const_mul 2).div hθm hθ.ne' using 1 |
| simp |
| have ha3 : Tendsto (fun j => (θ+η j)/(θ-η j)-1) atTop (𝓝 0) := by |
| simpa [hθ.ne',Pi.div_apply] using (hθp.div hθm hθ.ne').sub (tendsto_const_nhds (x:=1)) |
| simpa [a] using (ha1.add ha2).add ha3 |
| have hb : Tendsto b atTop (𝓝 (2*τ/θ)) := by |
| have hb1 := (tendsto_const_nhds (x:=2*τ)).div hθm hθ.ne' |
| have hb2 : Tendsto (fun j => 2-2*((θ-η j)/(θ+η j/8))) atTop (𝓝 0) := by |
| simpa [hθ.ne',Pi.div_apply] using (tendsto_const_nhds (x:= (2:ℝ))).sub ((hθm.div hθp8 hθ.ne').const_mul 2) |
| simpa [b,Pi.div_apply] using hb1.add hb2 |
| have he : Tendsto e atTop (𝓝 (32*Real.exp (-H/2))) := by |
| simpa [e] using ((hlamT.const_mul 4).mul_const H).add (tendsto_const_nhds (x:=32*Real.exp (-H/2))) |
| have hlt : 2*τ/θ+32*Real.exp (-H/2)<ε := by linarith |
| have hsum := (ha.add hb |>.add he).eventually (eventually_lt_nhds (by simpa using hlt)) |
| filter_upwards [hbound τ H hτ hτ1 hH,hsum,hη.eventually (eventually_lt_nhds hθ), |
| (Real.tendsto_log_atTop.comp hn).eventually (eventually_gt_atTop (0:ℝ))] with j hj hjlt hjη hjlog |
| have ha0 : 0 ≤ a j := by |
| have hr : 1 ≤ (θ+η j)/(θ-η j) := (one_le_div (sub_pos.mpr hjη)).mpr (by linarith [hη0 j]) |
| dsimp [a] |
| positivity |
| have hb0 : 0 ≤ b j := by |
| have hr : (θ-η j)/(θ+η j/8) ≤ 1 := (div_le_one (by linarith [hη0 j] : 0<θ+η j/8)).mpr (by linarith [hη0 j]) |
| have hr0 : 0 ≤ 2-2*((θ-η j)/(θ+η j/8)) := by linarith |
| dsimp [b] |
| positivity |
| have he0 : 0 ≤ e j := by dsimp [e]; positivity |
| have hden : 0 < U j+A j+1 := by linarith [hU j,hA j] |
| have hupper : D j ≤ (a j+b j+e j)*(U j+A j+1) := by |
| have hj' : D j ≤ a j*U j+b j*A j+e j := by |
| simpa only [a,b,e,add_assoc] using hj |
| nlinarith only [hj',hU j,hA j,ha0,hb0,he0,mul_nonneg ha0 (hA j),mul_nonneg hb0 (hU j),mul_nonneg he0 (hU j),mul_nonneg he0 (hA j)] |
| exact (div_le_iff₀ hden).mpr hupper |>.trans_lt hjlt |
| end WeakPinned |
|
|
| namespace WeakPinned |
| open Filter |
| open scoped Topology |
| open Global |
| variable (V : ℕ → Type u1) (I : ℕ → Type u2) (K : ℕ → Type u3) [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)] |
| [∀ j, Field (K j)] [∀ j, NumberField (K j)] |
| variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ) |
| (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j)) |
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| /-- Application of the scalar rate to the genuinely prepared arithmetic graphs, |
| with no degree bound on the sequence of number fields. -/ |
| theorem arithmetic_small_overlap_rate (hθ : 0<θ) |
| (hn : Tendsto (fun j => (Fintype.card (V j) : ℝ)) atTop atTop) |
| (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) |
| (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j)) |
| (t : ∀ j, I j → K j) |
| (ht : ∀ j e y, y ∈ (G j).targets e → |
| (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) : |
| Tendsto (fun j => smallOverlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) c / |
| (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) |
| atTop (𝓝 0) := by |
| apply small_overlap_rate (fun j => (Fintype.card (V j):ℝ)) η _ _ _ θ hθ hn hη hη0 |
| (fun j => overlapScale_nonneg _ _ _ _) (fun j => potentialScale_nonneg _ _ _ _) |
| (fun j => smallOverlapScale_nonneg _ _ _ _ c) |
| intro τ H hτ hτ1 hH |
| filter_upwards [hη.eventually (eventually_lt_nhds hθ), |
| (Real.tendsto_log_atTop.comp hn).eventually (eventually_ge_atTop (2:ℝ)), |
| hn.eventually (eventually_ge_atTop (2/τ))] with j hjη hjlog hjn |
| have hlog : 0 ≤ 1/Real.log (Fintype.card (V j):ℝ) := by positivity |
| have hlog2 : 1/Real.log (Fintype.card (V j):ℝ) ≤ 1/2 := by |
| exact one_div_le_one_div_of_le (by norm_num) hjlog |
| have hτn : 2≤τ*(Fintype.card (V j):ℝ) := by |
| have := (div_le_iff₀ hτ).mp hjn |
| nlinarith |
| exact (hG j).arithmetic_small_overlap (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) hjη (hη0 j) |
| (t j) (ht j) τ H hlog hlog2 hτ hτ1 hτn hH |
|
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| end WeakPinned |
|
|
| namespace WeakPinned.Core.Fibers |
| open scoped BigOperators |
| open Variance |
| variable {V : Type u4} {I : Type u5} [Fintype V] [Fintype I] [DecidableEq V] |
| variable (G : Fibers V I) |
|
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| def retainedWeight (T : ℝ) (y : V) : ℝ := if T≤((G.incoming y).card:ℝ) then G.incomingWeight y else 0 |
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| def retainedSet (T : ℝ) (y : V) : Finset V := if T≤((G.incoming y).card:ℝ) then G.incoming y else Finset.univ |
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| omit [Fintype I] in |
| lemma retainedWeight_nonneg (T : ℝ) (y : V) : 0≤G.retainedWeight T y := by |
| unfold retainedWeight incomingWeight edgeCount |
| split_ifs <;> positivity |
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| omit [Fintype I] in |
| lemma retainedWeight_le (T : ℝ) (y : V) : G.retainedWeight T y≤G.incomingWeight y := by |
| unfold retainedWeight |
| split_ifs |
| · exact le_rfl |
| · unfold incomingWeight edgeCount; positivity |
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| omit [Fintype I] in |
| lemma sum_retainedWeight (T : ℝ) (hG : 0<G.edgeCount) : (∑ y, G.retainedWeight T y)≤1 := by |
| rw [← G.sum_incomingWeight hG] |
| exact Finset.sum_le_sum (fun y _ => G.retainedWeight_le T y) |
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| omit [Fintype I] in |
| lemma retainedSet_size (T : ℝ) (hT : T≤Fintype.card V) (y : V) : T≤((G.retainedSet T y).card:ℝ) := by |
| unfold retainedSet |
| split_ifs with h |
| · exact h |
| · simpa using hT |
|
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| omit [Fintype I] in |
| lemma retained_domination (T M : ℝ) (hdom : ∀ x y, G.edgeLaw x y≤M*pairProb Finset.univ x y) |
| (hn : 2≤Fintype.card V) (x : V) : |
| (∑ y, G.retainedWeight T y*prob (G.retainedSet T y) x)≤M*prob Finset.univ x := by |
| refine le_trans (Finset.sum_le_sum (fun y _ => ?_)) (G.incoming_domination M hdom hn x) |
| unfold retainedWeight retainedSet |
| split_ifs |
| · exact le_rfl |
| · simp only [zero_mul] |
| unfold incomingWeight edgeCount prob |
| split_ifs <;> positivity |
| end WeakPinned.Core.Fibers |
|
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| namespace WeakPinned |
| open Filter Variance Global |
| open scoped Topology BigOperators |
| variable {V : ℕ → Type u6} {I : ℕ → Type u7} {K : ℕ → Type u8} [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)] |
| [∀ j, Field (K j)] [∀ j, NumberField (K j)] |
| variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ) |
| (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j)) |
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| lemma PreparedGraph.uniform_domination {V : Type u9} {I : Type u10} [Fintype V] [Fintype I] [DecidableEq V] |
| {G : Core.Fibers V I} {θ γ c η : ℝ} (h : PreparedGraph V I G θ γ c η) (hθ : 0<θ) |
| (hη : η≤θ/2) (x y : V) : G.edgeLaw x y≤(2/θ)*pairProb Finset.univ x y := by |
| have hh : η<θ := by linarith |
| refine (h.edge_domination hh x y).trans (mul_le_mul_of_nonneg_right ?_ (by |
| unfold pairProb |
| split_ifs |
| · have hn : (2:ℝ)≤Fintype.card V := by exact_mod_cast h.card_ge |
| simp only [Finset.card_univ] |
| exact div_nonneg (by norm_num) (mul_nonneg (by positivity) (by linarith)) |
| · exact le_rfl)) |
| apply (div_le_div_iff₀ (by linarith : 0<θ-η) hθ).mpr |
| linarith |
|
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| def fiberVariance (j : ℕ) : ℝ := weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)) |
| (varianceCell (G j).fiberWeight (G j).targets ((Fintype.card (V j):ℝ)^(1-c))) |
|
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| def incomingVariance (j : ℕ) : ℝ := weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)) |
| (varianceCell ((G j).retainedWeight ((Fintype.card (V j):ℝ)^γ)) |
| ((G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) ((Fintype.card (V j):ℝ)^(1-c))) |
|
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| /-- Both source sampling families satisfy the proved vanishing arithmetic |
| variance estimate, on every sequence of prepared models. No uniform degree |
| bound on the number fields is used. -/ |
| theorem sampling_variance_rates (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hcγ : c<γ) |
| (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop) |
| (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) (hηb : ∀ j, η j≤θ/2) |
| (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j)) |
| (t : ∀ j, I j → K j) (ht : ∀ j e y, y ∈ (G j).targets e → |
| (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) : |
| Tendsto (fun j => fiberVariance G c Z₁ Z₂ h₁ h₂ j / |
| (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) ∧ |
| Tendsto (fun j => incomingVariance G γ c Z₁ Z₂ h₁ h₂ j / |
| (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) := by |
| have hsmall := arithmetic_small_overlap_rate V I K G θ γ c η Z₁ Z₂ h₁ h₂ hθ hn hη hη0 hG t ht |
| have hηlt (j) : η j<θ := by linarith [hηb j] |
| have hpos (j) : 0<(G j).edgeCount := (hG j).edgeCount_pos (hηlt j) |
| have hdom (j x y) := (hG j).uniform_domination hθ (hηb j) x y |
| constructor |
| · exact arithmetic_variance_rate Z₁ Z₂ h₁ h₂ (fun j => (G j).fiberWeight) (fun j => (G j).targets) |
| γ c (2/θ) hγ hγ1 hcγ (hcγ.trans hγ1) (by positivity) hn |
| (fun j => (G j).fiberWeight_nonneg) (fun j => (G j).sum_fiberWeight (hpos j) |>.le) |
| (Eventually.of_forall fun j e => (hG j).fiber e |>.le) |
| (Eventually.of_forall fun j x => (G j).fiber_domination _ (hdom j) (hG j).card_ge x) hsmall |
| · apply arithmetic_variance_rate Z₁ Z₂ h₁ h₂ |
| (fun j => (G j).retainedWeight ((Fintype.card (V j):ℝ)^γ)) |
| (fun j => (G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) γ c (2/θ) |
| hγ hγ1 hcγ (hcγ.trans hγ1) (by positivity) hn |
| (fun j y => (G j).retainedWeight_nonneg _ y) |
| (fun j => (G j).sum_retainedWeight _ (hpos j)) _ _ hsmall |
| · apply Eventually.of_forall |
| intro j y |
| apply (G j).retainedSet_size |
| have hn1 : (1:ℝ)≤Fintype.card (V j) := by exact_mod_cast (by have := (hG j).card_ge; omega : 1≤Fintype.card (V j)) |
| simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn1 hγ1.le |
| · exact Eventually.of_forall fun j x => (G j).retained_domination _ _ (hdom j) (hG j).card_ge x |
|
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| end WeakPinned |
|
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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