| import Mathlib |
| import OAI.Probability.Perceptron.Variational.ConvexUniform |
| import OAI.Probability.Perceptron.Variational.GlobalContact |
| import OAI.Probability.Perceptron.Pressure.PatternFree |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma perturbationScale_tendsto_zero : |
| Tendsto (fun n : ℕ => perturbationScale (n+1)) atTop (𝓝 0) := by |
| unfold perturbationScale |
| convert (tendsto_rpow_neg_atTop (by norm_num : (0:ℝ)<1/16)).comp |
| (tendsto_natCast_atTop_atTop.comp (tendsto_add_atTop_nat 1)) using 1 |
| simp only [neg_div] |
| rfl |
|
|
| lemma sourcePatternFree_uniform_upper {k : ℕ} (w : Fin (k+1)→ℝ) |
| (hw : ∀ i, 0<w i) (hw1 : ∑ i, w i=1) (H : ℝ) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ n : ℕ in atTop, ∀ (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ), h∈sourceStepFieldCap k H → |
| ∀ u : Fin (n+1)→ℝ, (∀ j, u j∈Icc 0 2) → |
| ∀ q : Fin (k+1)→Time, Monotone q → (q (Fin.last k):ℝ)<1 → |
| sourceExpectedPressure n k f p d h (stepCumulative w) 0 u ≤ |
| finiteSphericalDualObjective w h (fun i => (hw i).le) hw1 q+ε := by |
| have hs := finiteSphericalFieldValue_compact_uniform w hw hw1 |
| (sourceStepFieldCap_compact k H) (fun h hh => ⟨hh.2,hh.1.1 0⟩) |
| rw [Metric.tendstoUniformlyOn_iff] at hs |
| have ht : Tendsto (fun n : ℕ => 8*perturbationScale (n+1)^2) atTop (𝓝 0) := by |
| simpa using perturbationScale_tendsto_zero.pow 2 |>.const_mul 8 |
| filter_upwards [hs (ε/2) (by positivity),ht.eventually (gt_mem_nhds (show (0:ℝ)<ε/2 by positivity))] |
| with n hn hn' f p d h hh u hu q hq hq1 |
| have hc := sourceExpectedPressure_perturbation_comparison n k f p d h hh.1.1 hh.2 |
| (stepCumulative w) (stepCumulative_strictMono w hw) (stepCumulative_pos w hw) |
| (stepCumulative_lt_one w hw hw1) 0 (u := fun _ => 0) (v := u) |
| (fun j => ⟨le_rfl,by norm_num⟩) hu |
| change |sourceExpectedPressure n k f p d h (stepCumulative w) 0 u- |
| sourceExpectedPressure n k f p d h (stepCumulative w) 0 (fun _ => 0)|≤_ at hc |
| rw [sourceExpectedPressure_patternFree n k f p d w h hw hw1 hh.2 (hh.1.1 0)] at hc |
| obtain ⟨r,hr,hr1,hrmin,-⟩ := finiteSphericalFieldValue_pointwise_dual w h hw hw1 hh.2 (hh.1.1 0) |
| have hle := hrmin.2 (show finiteSphericalDualObjective w h (fun i => (hw i).le) hw1 q∈ |
| finiteSphericalDualValues w h (fun i => (hw i).le) hw1 from ⟨q,⟨hq,hq1⟩,rfl⟩) |
| have he := hn h hh |
| rw [Real.dist_eq,hrmin.csInf_eq] at he |
| have hab := (abs_le.mp hc).2 |
| have hae := (abs_lt.mp he).1 |
| linarith |
|
|
| lemma sourcePressure_uniform_upper {k : ℕ} (w : Fin (k+1)→ℝ) |
| (hw : ∀ i, 0<w i) (hw1 : ∑ i, w i=1) (H : ℝ) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ n : ℕ in atTop, ∀ (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ), h∈sourceStepFieldCap k H → |
| ∀ u : Fin (n+1)→ℝ, (∀ j, u j∈Icc 0 2) → |
| ∀ q : Fin (k+1)→Time, Monotone q → (q (Fin.last k):ℝ)<1 → ∀ t : ℝ≥0, |
| sourceExpectedPressure n k f p d h (stepCumulative w) t u ≤ |
| finiteSphericalDualObjective w h (fun i => (hw i).le) hw1 q+ε+(t:ℝ)*‖f‖ := by |
| filter_upwards [sourcePatternFree_uniform_upper w hw hw1 H hε] with n hn |
| intro f p d h hh u hu q hq hq1 t |
| have hc := sourceExpectedPressure_density_comparison n k f p d h (stepCumulative w) u |
| (stepCumulative_strictMono w hw) (stepCumulative_pos w hw) |
| (stepCumulative_lt_one w hw hw1) 0 t |
| rw [NNReal.coe_zero,sub_zero,abs_of_nonneg t.coe_nonneg] at hc |
| linarith [(abs_le.mp hc).2,hn f p d h hh u hu q hq hq1] |
|
|
| lemma sourceContactObjective_zero_lower {k : ℕ} (w : Fin (k+1)→ℝ) |
| (hw : ∀ i, 0<w i) (hw1 : ∑ i, w i=1) (H : ℝ) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ n : ℕ in atTop, ∀ (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (q : Fin (k+1)→Time), Monotone q → (q (Fin.last k):ℝ)<1 → |
| ∀ δ α (a : SourceContactParameter n k), a∈sourceContactCompactDomain n k α H → a.1=0 → |
| -ε ≤ sourceStepContactObjective n k P f p d w q (fun i => (hw i).le) hw1 δ a := by |
| filter_upwards [sourcePatternFree_uniform_upper w hw hw1 H hε] with n hn |
| intro P f p d q hq hq1 δ α a ha ht |
| have he := hn f p d a.2.1 ha.2.1 a.2.2 |
| (fun j => ⟨by linarith [ha.2.2.1 j],ha.2.2.2 j⟩) q hq hq1 |
| have hpen : 0≤quadraticBoxPenalty (sourcePenaltyWeight (n+1)) a.2.2 := by |
| unfold quadraticBoxPenalty sourcePenaltyWeight |
| positivity |
| have hsum : (∑ i, w i*(q i:ℝ)*a.2.1 i)=(∑ i, w i*a.2.1 i*(q i:ℝ)) := by |
| apply Finset.sum_congr rfl |
| intros |
| ring |
| unfold sourceStepContactObjective sourceJointPressure |
| rw [ht,NNReal.coe_zero,zero_mul,add_zero,hsum] |
| unfold finiteSphericalDualObjective at he |
| linarith |
|
|
| lemma sourceNegativeContact_positive_density {k : ℕ} (w : Fin (k+1)→ℝ) |
| (hw : ∀ i, 0<w i) (hw1 : ∑ i, w i=1) (H : ℝ) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ n : ℕ in atTop, ∀ (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (q : Fin (k+1)→Time), Monotone q → (q (Fin.last k):ℝ)<1 → |
| ∀ δ α (a : SourceContactParameter n k), a∈sourceContactCompactDomain n k α H → |
| sourceStepContactObjective n k P f p d w q (fun i => (hw i).le) hw1 δ a < -ε → 0<a.1 := by |
| filter_upwards [sourceContactObjective_zero_lower w hw hw1 H hε] with n hn |
| intro P f p d q hq hq1 δ α a ha hneg |
| by_contra ht |
| have he : a.1=0 := le_antisymm (le_of_not_gt ht) zero_le |
| exact (not_le_of_gt hneg) (hn P f p d q hq hq1 δ α a ha he) |
|
|
| lemma sourceContactObjective_uniform_lower {k : ℕ} (w : Fin (k+1)→ℝ) |
| (hw : ∀ i, 0<w i) (hw1 : ∑ i, w i=1) (H : ℝ) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ n : ℕ in atTop, ∀ (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (q r : Fin (k+1)→Time), Monotone r → (r (Fin.last k):ℝ)<1 → |
| ∀ δ α (a : SourceContactParameter n k), a∈sourceContactCompactDomain n k α H → |
| (∑ i, w i*((r i:ℝ)-(q i:ℝ))*a.2.1 i)+ |
| (entropy (weightedStepTrial w q (fun i => (hw i).le) hw1)).toReal- |
| (entropy (weightedStepTrial w r (fun i => (hw i).le) hw1)).toReal+ |
| (a.1:ℝ)*(controlValue P f (weightedStepTrial w q (fun i => (hw i).le) hw1)+δ-‖f‖)-ε ≤ |
| sourceStepContactObjective n k P f p d w q (fun i => (hw i).le) hw1 δ a := by |
| filter_upwards [sourcePressure_uniform_upper w hw hw1 H hε] with n hn |
| intro P f p d q r hr hr1 δ α a ha |
| have he := hn f p d a.2.1 ha.2.1 a.2.2 |
| (fun j => ⟨by linarith [ha.2.2.1 j],ha.2.2.2 j⟩) r hr hr1 a.1 |
| have hpen : 0≤quadraticBoxPenalty (sourcePenaltyWeight (n+1)) a.2.2 := by |
| unfold quadraticBoxPenalty sourcePenaltyWeight |
| positivity |
| have hsum : (∑ i, w i*((r i:ℝ)-(q i:ℝ))*a.2.1 i)= |
| (∑ i, w i*a.2.1 i*(r i:ℝ))-(∑ i, w i*(q i:ℝ)*a.2.1 i) := by |
| rw [← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intros |
| ring |
| unfold sourceStepContactObjective sourceJointPressure |
| unfold finiteSphericalDualObjective at he |
| rw [hsum] |
| linarith |
|
|
| lemma sourceNegativeContact_field_interior {k : ℕ} (w : Fin (k+1)→ℝ) |
| (hw : ∀ i, 0<w i) (hw1 : ∑ i, w i=1) |
| (P : Measure BrownianPath) [IsProbabilityMeasure P] (f : ℝ →ᵇ ℝ) |
| (q r : Fin (k+1)→Time) (hr : Monotone r) (hr1 : (r (Fin.last k):ℝ)<1) |
| {η : ℝ} (hη : 0<η) (hqr : ∀ i, (q i:ℝ)+η≤(r i:ℝ)) (α : ℝ≥0) |
| (H : ℝ) (hH : (entropy (weightedStepTrial w r (fun i => (hw i).le) hw1)).toReal+ |
| 2*(α:ℝ)*‖f‖ ≤ η*w (Fin.last k)*H) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ n : ℕ in atTop, ∀ (p d : Fin (n+1)→ℕ) (δ : ℝ), 0≤δ → |
| ∀ a : SourceContactParameter n k, a∈sourceContactCompactDomain n k α H → |
| sourceStepContactObjective n k P f p d w q (fun i => (hw i).le) hw1 δ a < -ε → |
| a.2.1 (Fin.last k)<H := by |
| filter_upwards [sourceContactObjective_uniform_lower w hw hw1 H hε] with n hn |
| intro p d δ hδ a ha hneg |
| have he := hn P f p d q r hr hr1 δ α a ha |
| have hs : (entropy (weightedStepTrial w q (fun i => (hw i).le) hw1)).toReal≥0 := ENNReal.toReal_nonneg |
| have hv := neg_norm_le_controlValue P f (weightedStepTrial w q (fun i => (hw i).le) hw1) |
| have ht : -2*(α:ℝ)*‖f‖ ≤ |
| (a.1:ℝ)*(controlValue P f (weightedStepTrial w q (fun i => (hw i).le) hw1)+δ-‖f‖) := by |
| have hm := mul_le_mul_of_nonneg_left (show -2*‖f‖ ≤ |
| controlValue P f (weightedStepTrial w q (fun i => (hw i).le) hw1)+δ-‖f‖ by linarith) a.1.coe_nonneg |
| have ha' : (a.1:ℝ)≤(α:ℝ) := ha.1.2 |
| nlinarith [mul_le_mul_of_nonneg_right ha' (norm_nonneg f)] |
| have hsum : η*w (Fin.last k)*a.2.1 (Fin.last k) ≤ |
| ∑ i, w i*((r i:ℝ)-(q i:ℝ))*a.2.1 i := by |
| apply le_trans _ (Finset.single_le_sum (fun i (_ : i∈Finset.univ) => |
| mul_nonneg (mul_nonneg (hw i).le (by linarith [hqr i])) (ha.2.1.1.1 i)) (Finset.mem_univ (Fin.last k))) |
| nlinarith [mul_le_mul_of_nonneg_left (hqr (Fin.last k)) |
| (mul_nonneg (hw (Fin.last k)).le (ha.2.1.1.1 (Fin.last k)))] |
| by_contra hnot |
| have hEq : a.2.1 (Fin.last k)=H := le_antisymm (ha.2.1.1.2 (Fin.last k)) (le_of_not_gt hnot) |
| rw [hEq] at hsum |
| linarith |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
|
| end OAI |
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|
