| import Mathlib |
| import OAI.Probability.SKRatio.Dynamics.Median |
| import OAI.Probability.SKRatio.Dynamics.PoissonBinomial |
| import OAI.Probability.SKRatio.Dynamics.ClockBridge |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology Matrix |
| open ContinuousLinearMap MeasureTheory Filter ProbabilityTheory SKRatioClock |
| namespace SKRatio.Calculus |
|
|
| lemma median_attempts_diverge_of_log_lower |
| (G : ∀ n : ℕ, Set (Disorder n)) {c : ℝ} (hc : 0 < c) |
| (hscale : ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, c*Real.log n ≤ medianTime g) : |
| ∀ R : ℝ, ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, R ≤ (n:ℝ)*medianTime g := by |
| have hlog : Tendsto (fun n : ℕ => Real.log (n:ℝ)) atTop atTop := |
| Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop |
| have hlim : Tendsto (fun n : ℕ => (n:ℝ)*(c*Real.log n)) atTop atTop := |
| tendsto_natCast_atTop_atTop.atTop_mul_atTop₀ (hlog.const_mul_atTop hc) |
| intro R |
| filter_upwards [hscale,hlim.eventually_ge_atTop R] with n hn hR |
| intro g hg |
| exact hR.trans (mul_le_mul_of_nonneg_left (hn g hg) (Nat.cast_nonneg n)) |
|
|
| lemma upper_median_clock_diverges |
| (G : ∀ n : ℕ, Set (Disorder n)) {b : ℝ} (hb : 0 < b) |
| (hdiv : ∀ R : ℝ, ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, R ≤ (n:ℝ)*medianTime g) : |
| ∀ K : ℕ, ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, |
| K ≤ ⌈(1+b)*(n:ℝ)*medianTime g⌉₊ := by |
| intro K |
| have hp : 0 < 1+b := by linarith |
| filter_upwards [hdiv ((K:ℝ)/(1+b))] with n hn |
| intro g hg |
| have hx := (div_le_iff₀ hp).mp (hn g hg) |
| have hy : (K:ℝ) ≤ (⌈(1+b)*(n:ℝ)*medianTime g⌉₊:ℝ) := by |
| apply le_trans _ (Nat.le_ceil _) |
| nlinarith only [hx] |
| exact_mod_cast hy |
|
|
| lemma upper_clock_shifted_time {n : ℕ} (hn : 0 < n) (t : ℝ) |
| {b q y : ℝ} (hb : 0 < b) (hq : 0 < q) |
| (hy : |y| ≤ (q*b/(2*(1+b)))*Real.sqrt (⌈(1+b)*(n:ℝ)*t⌉₊:ℝ)) : |
| (1+b/2)*t ≤ |
| (q*(⌈(1+b)*(n:ℝ)*t⌉₊:ℝ)+y*Real.sqrt (⌈(1+b)*(n:ℝ)*t⌉₊:ℝ))/(q*n) := by |
| let M : ℕ := ⌈(1+b)*(n:ℝ)*t⌉₊ |
| let c := q*b/(2*(1+b)) |
| have hc : 0 < c := by dsimp [c]; positivity |
| have hcq : c < q := by |
| dsimp [c] |
| apply (div_lt_iff₀ (by positivity : 0 < 2*(1+b))).mpr |
| nlinarith |
| have hn' : (0:ℝ) < n := Nat.cast_pos.mpr hn |
| have hM : (1+b)*(n:ℝ)*t ≤ (M:ℝ) := Nat.le_ceil _ |
| have hsq : Real.sqrt (M:ℝ)^2 = (M:ℝ) := Real.sq_sqrt (Nat.cast_nonneg _) |
| have herr : -c*(M:ℝ) ≤ y*Real.sqrt (M:ℝ) := by |
| have hh := mul_le_mul_of_nonneg_right hy (Real.sqrt_nonneg (M:ℝ)) |
| have hy' := mul_le_mul_of_nonneg_right (neg_abs_le y) (Real.sqrt_nonneg (M:ℝ)) |
| change |y| * Real.sqrt (M:ℝ) ≤ c*Real.sqrt (M:ℝ)*Real.sqrt (M:ℝ) at hh |
| nlinarith |
| apply (le_div_iff₀ (mul_pos hq hn')).mpr |
| have hmain := mul_le_mul_of_nonneg_left hM (sub_pos.mpr hcq).le |
| have hid : (q-c)*(1+b) = q*(1+b/2) := by |
| dsimp [c] |
| field_simp |
| ring |
| change _ ≤ q*(M:ℝ)+y*Real.sqrt (M:ℝ) |
| calc |
| _ = (q-c)*((1+b)*(n:ℝ)*t) := by |
| calc |
| _ = ((q-c)*(1+b))*(n:ℝ)*t := by rw [hid]; ring |
| _ = _ := by ring |
| _ ≤ (q-c)*(M:ℝ) := hmain |
| _ ≤ _ := by nlinarith |
|
|
| lemma uniform_upper_clock_shift |
| (G : ∀ n : ℕ, Set (Disorder n)) {b q : ℝ} (hb : 0 < b) (hq : 0 < q) |
| (hdiv : ∀ R : ℝ, ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, R ≤ (n:ℝ)*medianTime g) |
| (y : ℝ) : |
| ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, |
| (1+b/2)*medianTime g ≤ |
| (q*(⌈(1+b)*(n:ℝ)*medianTime g⌉₊:ℝ)+ |
| y*Real.sqrt (⌈(1+b)*(n:ℝ)*medianTime g⌉₊:ℝ))/(q*n) := by |
| have hc : 0 < q*b/(2*(1+b)) := by positivity |
| have hs := (Real.tendsto_sqrt_atTop.comp |
| (tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n:ℝ)) atTop atTop)).const_mul_atTop hc |
| obtain ⟨N,hN⟩ := eventually_atTop.mp (hs.eventually_ge_atTop |y|) |
| filter_upwards [upper_median_clock_diverges G hb hdiv N,eventually_gt_atTop 0] with n hn hn0 |
| intro g hg |
| exact upper_clock_shifted_time hn0 (medianTime g) hb hq (hN _ (hn g hg)) |
|
|
| theorem uniform_discrete_upper_of_holding |
| (G : ∀ n : ℕ, Set (Disorder n)) {b q : ℝ} |
| (hb : 0 < b) (hq : 0 < q) (hq1 : q < 1) |
| (hdiv : ∀ R : ℝ, ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, R ≤ (n:ℝ)*medianTime g) |
| (hhold : ∀ n, 0 < n → ∀ g ∈ G n, ∀ x, 1-q ≤ transition g x x) |
| (hcont : ∀ δ : ℝ, 0 < δ → ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, |
| continuousDistance g ((1+b/2)*medianTime g) ≤ δ) : |
| ∀ δ : ℝ, 0 < δ → ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, |
| discreteDistance g ⌈(1+b)*(n:ℝ)*medianTime g⌉₊ ≤ δ := by |
| classical |
| let ι : ℕ → Type := fun n => Option |
| ({g : Disorder n // g ∈ G n ∧ 0 < n} × Spin n × |
| {f : Observables n // ∀ x, |f x| ≤ 1}) |
| let M : (n : ℕ) → ι n → ℕ := fun n i => match i with |
| | none => n+1 |
| | some j => ⌈(1+b)*(n:ℝ)*medianTime j.1.val⌉₊ |
| let a : (n : ℕ) → ι n → ℕ → ℝ := fun _ i k => match i with |
| | none => 0 |
| | some j => (residualAttempt (coupling j.1.val) q ^ k) j.2.2.val j.2.1- |
| FiniteLaw.mean (mass j.1.val 0) j.2.2.val |
| have hM : ∀ K : ℕ, ∀ᶠ n in atTop, ∀ i : ι n, K ≤ M n i := by |
| intro K |
| filter_upwards [upper_median_clock_diverges G hb hdiv K,eventually_ge_atTop K] with n hn hnK |
| intro i |
| cases i with |
| | none => exact hnK.trans (Nat.le_succ n) |
| | some j => exact hn j.1.val j.1.property.1 |
| have hab : ∀ n (i : ι n) k, |a n i k| ≤ 2 := by |
| intro n i k |
| cases i with |
| | none => norm_num [a] |
| | some j => |
| exact residual_test_bound j.1.val hq |
| (hhold n j.1.property.2 j.1.val j.1.property.1) j.2.2.val j.2.2.property j.2.1 k |
| have hp : ∀ y ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ i : ι n, |
| |∑' j, Clock.poissonMass (q*M n i+y*Real.sqrt (M n i)) j*a n i j| < ε := by |
| intro y ε hε |
| filter_upwards [uniform_upper_clock_shift G hb hq hdiv y,hcont (ε/4) (by positivity)] with n hn hcn |
| intro i |
| cases i with |
| | none => simpa only [a,mul_zero,tsum_zero,abs_zero] using hε |
| | some j => |
| change |∑' k, Clock.poissonMass |
| (q*(⌈(1+b)*(n:ℝ)*medianTime j.1.val⌉₊:ℝ)+ |
| y*Real.sqrt (⌈(1+b)*(n:ℝ)*medianTime j.1.val⌉₊:ℝ)) k* |
| ((residualAttempt (coupling j.1.val) q ^ k) j.2.2.val j.2.1- |
| FiniteLaw.mean (mass j.1.val 0) j.2.2.val)| < ε |
| rw [residual_poisson_test j.1.property.2 j.1.val hq.ne'] |
| have htime := hn j.1.val j.1.property.1 |
| have ht : 0 ≤ (1+b/2)*medianTime j.1.val := |
| mul_nonneg (by linarith) (medianTime_mem j.1.val).1 |
| have hbd := (continuous_test_bound j.1.val _ j.2.2.val j.2.2.property j.2.1).trans |
| (mul_le_mul_of_nonneg_left (continuousDistance_antitone j.1.val ht htime) (by norm_num)) |
| have hc := hcn j.1.val j.1.property.1 |
| linarith |
| have hbin := Clock.poisson_to_binomial_uniform_clocks hq hq1 M hM a hab hp |
| intro δ hδ |
| filter_upwards [hbin δ hδ,eventually_gt_atTop 0] with n hn hn0 |
| intro g hg |
| apply Finset.sup'_le |
| intro x _ |
| apply FiniteLaw.totalVariation_le_of_tests |
| intro f hf |
| let i : ι n := some (⟨g,hg,hn0⟩,x,⟨f,hf⟩) |
| have hbd := hn i |
| change |∑' k, Clock.binomialMass q ⌈(1+b)*(n:ℝ)*medianTime g⌉₊ k* |
| ((residualAttempt (coupling g) q ^ k) f x-FiniteLaw.mean (mass g 0) f)| < δ at hbd |
| rw [residual_binomial_test g hq.ne'] at hbd |
| change |((transition g ^ ⌈(1+b)*(n:ℝ)*medianTime g⌉₊) *ᵥ f) x- |
| FiniteLaw.mean (mass g 0) f| ≤ 2*δ |
| linarith |
|
|
| end SKRatio.Calculus |
|
|
| end |
| end |
|
|
| end OAI |
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|
