| import OAI.Probability.DirectionalWalk.EntropyUpperBound |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
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| namespace DirectionalZeroOne |
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| open scoped Classical |
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| lemma dyadic_entropy_contradiction (p a : ℕ → ℝ) |
| (hp : ∀ j, 0 ≤ p j) (ha : ∀ j, 0 ≤ a j) (hs : Summable a) |
| (P W R E A B : ℝ) (hP : 0 < P) (hW : 0 ≤ W) (hR : 0 ≤ R) |
| (hE : 0 ≤ E) (hA : 0 ≤ A) (hB : 0 ≤ B) |
| (hlower : ∀ (J G c l m : ℕ), 0 < G → G+c ≤ l → l < J → |
| (1/2 : ℝ)^G*a l ≤ (1/2 : ℝ)^m → |
| P ≤ W*(1/2 : ℝ)^c + (∑ j ∈ Finset.Icc (l-(G+c)) l, p j) + |
| R*(1/2 : ℝ)^J + E*J*(m+1)^2*(1/2 : ℝ)^m) |
| (hupper : ∀ J L : ℕ, (L*Real.log 2)*(∑ j ∈ Finset.range J, p j) ≤ |
| A*J+(L*Real.log 2)*(2:ℝ)^L*B) : False := by |
| let S := ∑' j, a j |
| let C := 4*S+1 |
| have hS0 : 0 ≤ S := tsum_nonneg ha |
| have hC : 0 < C := by dsimp [C];linarith |
| have hSC : 4*S ≤ C := by dsimp [C];linarith |
| have hS (J : ℕ) : ∑ j ∈ Finset.range J, a j ≤ S := |
| Summable.sum_le_tsum _ (fun j _ => ha j) hs |
| obtain ⟨c,hc⟩ := (eventually_dyadic_quadratic_lt W (P/4) (by positivity)).exists |
| have hcsmall : W*(1/2 : ℝ)^c < P/4 := |
| (mul_le_mul_of_nonneg_left (dyadic_decay_le_quadratic c) hW).trans_lt hc |
| have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| obtain ⟨K,hK⟩ := exists_nat_gt (12*(A+1)/(P*Real.log 2)) |
| have hk0 : (0:ℝ) ≤ 12*(A+1)/(P*Real.log 2) := by positivity |
| have hKpos : 0 < K := by exact_mod_cast (lt_of_le_of_lt hk0 hK) |
| have hKbig : 3*(A+1) < P*K*Real.log 2/4 := by |
| have hh := (div_lt_iff₀ (mul_pos hP hlog)).mp hK |
| nlinarith only [hh] |
| obtain ⟨t,htc,ht,hCt,hwJ,hr,herr,hcorr⟩ := |
| dyadic_large_parameters K c hKpos C R E B P hC hR hE hB hP |
| let J := (2:ℕ)^(2*K*t) |
| let w := 2*t+c |
| have hJpos : (0:ℝ) < J := by dsimp [J];positivity |
| have htpos : (0:ℝ) < t := by exact_mod_cast ht |
| have hw : (w:ℝ)+1 ≤ 3*t := by |
| have hh : (c:ℝ)+1 ≤ t := by exact_mod_cast htc |
| dsimp [w] |
| push_cast |
| linarith only [hh] |
| have hlabel (l : ℕ) (hwl : w ≤ l) (hlJ : l < J) (hgood : a l ≤ C/J) : |
| P/2 ≤ ∑ j ∈ Finset.Icc (l-w) l, p j := by |
| have hdelta : (1/2 : ℝ)^(2*t)*a l ≤ (1/2 : ℝ)^((2*K+1)*t) := by |
| calc |
| _ ≤ (1/2 : ℝ)^(2*t)*(C/J) := mul_le_mul_of_nonneg_left hgood (by positivity) |
| _ ≤ (1/2 : ℝ)^(2*t)*((2:ℝ)^t/J) := |
| mul_le_mul_of_nonneg_left (div_le_div_of_nonneg_right hCt hJpos.le) (by positivity) |
| _ = _ := by |
| dsimp [J] |
| push_cast |
| rw [dyadic_delta_identity] |
| congr 1 |
| ring |
| have hh := hlower J (2*t) c l ((2*K+1)*t) (by omega) hwl hlJ hdelta |
| change P ≤ W*(1/2:ℝ)^c + (∑ j ∈ Finset.Icc (l-w) l, p j) + R*(1/2:ℝ)^J + |
| E*J*((((2*K+1)*t : ℕ):ℝ)+1)^2*(1/2:ℝ)^((2*K+1)*t) at hh |
| change R*(1/2:ℝ)^J ≤ P/8 at hr |
| have herr' : E*J*((((2*K+1)*t : ℕ):ℝ)+1)^2*(1/2:ℝ)^((2*K+1)*t) ≤ P/8 := by |
| simpa only [J,Nat.cast_pow,Nat.cast_ofNat] using herr |
| linarith only [hh,hcsmall,hr,herr'] |
| have hc2 : 0 < P/2 := by positivity |
| have hcount := window_count_lower p a hp ha S C (P/2) hS hC hc2 hSC J w hwJ hlabel |
| have hu := hupper J (K*t) |
| have hu' : (((K*t : ℕ):ℝ)*Real.log 2)*(∑ j ∈ Finset.range J,p j) ≤ (A+1)*J := by |
| have hcorr' : ((K*t : ℕ):ℝ)*Real.log 2*(2:ℝ)^(K*t)*B ≤ (J:ℝ) := by |
| simpa only [J,Nat.cast_pow,Nat.cast_ofNat] using hcorr |
| linarith only [hu,hcorr'] |
| have hcoef : 0 ≤ ((K*t : ℕ):ℝ)*Real.log 2 := by positivity |
| have hAJ : 0 ≤ (A+1)*(J:ℝ) := by positivity |
| have hfinal : (P*K*Real.log 2/4)*(t:ℝ)*(J:ℝ) ≤ (3*(A+1))*(t:ℝ)*(J:ℝ) := by |
| calc |
| _ = (((K*t : ℕ):ℝ)*Real.log 2)*((P/2)*J/2) := by push_cast;ring |
| _ ≤ (((K*t : ℕ):ℝ)*Real.log 2)*((w+1)*(∑ j ∈ Finset.range J,p j)) := |
| mul_le_mul_of_nonneg_left hcount hcoef |
| _ = (w+1)*((((K*t : ℕ):ℝ)*Real.log 2)*(∑ j ∈ Finset.range J,p j)) := by ring |
| _ ≤ (w+1)*((A+1)*J) := mul_le_mul_of_nonneg_left hu' (by positivity) |
| _ ≤ (3*t)*((A+1)*J) := mul_le_mul_of_nonneg_right hw hAJ |
| _ = _ := by ring |
| have hf := le_of_mul_le_mul_right (le_of_mul_le_mul_right hfinal hJpos) htpos |
| exact not_le_of_gt hKbig hf |
|
|
| lemma not_both_axis_nonBacktracking {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (hell : StrictEllipticity μ) (e : Step d) |
| (hp : ∀ b, 0 < annealed μ 0 (nonBacktracking (axisDirection (placedAxis e b)))) : False := by |
| let : ∀ b, IsProbabilityMeasure (slabLaw μ (axisDirection (placedAxis e b))) := |
| fun b => slabLaw_probability μ _ (hp b) |
| let ν := fun b => slabLaw μ (axisDirection (placedAxis e b)) |
| let p (j : ℕ) := (twoTapeLaw ν).real (commonBandContact e (2^j)) |
| let a (l : ℕ) := (2:ℝ)^l*(axisWidthTail μ e (2^(l-1))).toReal |
| let P := (annealed μ 0).real (nonBacktracking (axisDirection e))^2 |
| let W := 4*(∫ Z, (firstCommonWidth (placedWidth e) Z : ℝ) ∂twoTapeLaw ν) |
| let R := ∫ a, slabRadius a ∂ν false |
| let E := ((annealed μ 0).real (nonBacktracking (axisDirection e)))⁻¹ * |
| ((annealed μ 0).real (nonBacktracking (axisDirection (oppositeStep e))))⁻¹ * |
| (24/posteriorExponent)*((d+1)*6) |
| have hpos : 0 < (annealed μ 0).real (nonBacktracking (axisDirection e)) := |
| ENNReal.toReal_pos (ne_of_gt (hp false)) (measure_ne_top _ _) |
| have hP : 0 < P := by dsimp [P];positivity |
| have hW : 0 ≤ W := mul_nonneg (by norm_num) (integral_nonneg (fun _ => Nat.cast_nonneg _)) |
| have hR : 0 ≤ R := integral_nonneg slabRadius_nonneg |
| have hE : 0 ≤ E := by dsimp [E];have := posteriorExponent_pos;positivity |
| obtain ⟨A,B,hA,hB,hupper⟩ := commonBand_entropy_upper μ hell e hp |
| apply dyadic_entropy_contradiction p a (fun _ => measureReal_nonneg) |
| (fun _ => mul_nonneg (by positivity) ENNReal.toReal_nonneg) |
| (summable_shifted_axisTail μ hell e (hp false)) P W R E A B hP hW hR hE hA hB _ hupper |
| intro J G c l m hG hl hlJ hΔ |
| have hh := commonBand_good_label μ hell e hp J G c l m hG hl hlJ hΔ |
| dsimp only at hh |
| dsimp only [P,W,R,E,p,ν] |
| nlinarith only [hh] |
|
|
| theorem directional_zero_one (d : ℕ) (hd : 3 ≤ d) |
| (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (hell : StrictEllipticity μ) (ℓ : Fin d → ℝ) (hℓ : ℓ ≠ 0) : |
| annealed μ 0 (escape ℓ) = 0 ∨ annealed μ 0 (escape ℓ) = 1 := by |
| cases d with |
| | zero => omega |
| | succ d => |
| by_contra hh |
| have hne0 : annealed μ 0 (escape ℓ) ≠ 0 := fun h => hh (Or.inl h) |
| have hne1 : annealed μ 0 (escape ℓ) ≠ 1 := fun h => hh (Or.inr h) |
| have h0 : 0 < annealed μ 0 (escape ℓ) := bot_lt_iff_ne_bot.mpr hne0 |
| have h1 : annealed μ 0 (escape ℓ) < 1 := lt_of_le_of_ne (prob_le_one) hne1 |
| obtain ⟨i,hi,hi'⟩ := mixed_direction_has_mixed_coordinate μ hell ℓ hℓ h0 h1 |
| apply not_both_axis_nonBacktracking μ hell (i,true) |
| intro b |
| cases b |
| · simpa only [placedAxis, Bool.false_eq_true,↓reduceIte,axisDirection] using |
| positive_nonBacktracking μ (Pi.single i 1) hi |
| · change 0 < annealed μ 0 (nonBacktracking (axisDirection (oppositeStep (i,true)))) |
| rw [axisDirection_opposite] |
| simpa only [axisDirection,↓reduceIte] using positive_nonBacktracking μ (-Pi.single i 1) hi' |
|
|
| end DirectionalZeroOne |
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| end OAI |
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