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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | import OAI.Probability.DirectionalWalk.EntropyUpperBound
namespace OAI
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped ENNReal BigOperators Topology
namespace DirectionalZeroOne
open scoped Classical
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
end OAI
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