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135052a | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | import Mathlib
import OAI.Probability.SKBarriers.Dynamics.GreedyStep
import OAI.Probability.SKBarriers.Dynamics.GreedyRecursion
namespace OAI
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem greedy_marker_windows {n : ℕ} (hn : 0 < n) (μ : ProbabilityMeasure ℝ)
(pool : ℕ → Finset (Config n)) (fallback : ℕ → Config n)
(hf : ∀ j, fallback j ∈ pool j) (base : ℕ → Config n) (x : ℕ → Config n)
(r : ℕ → ℝ) (B H p m : ℕ) (t ρ d b b' q : ℝ)
(hp : 0 < p) (hB : 0 < B) (hHB : H < B) (hH : ⌈2/b^2⌉₊ ≤ H)
(ht : 0 < t) (hρ : 0 < ρ) (hd : 0 ≤ d) (hb : 0 < b)
(hpack : 3*t < b^2/2) (hcsmall : 3*t < b) (hEst : 6*p*ρ^20 < t)
(hsmall' : 2*t+6*p*ρ^20+2*(n:ℝ)^(-(1:ℝ)/100) < b')
(hsep' : 12*p*ρ^20+4*(n:ℝ)^(-(1:ℝ)/100) < 3*ρ)
(hwidth : ρ^8 < t/2)
(hsmall : 6*p*ρ^20+2*(n:ℝ)^(-(1:ℝ)/100)+ρ^20+d/2 < ρ^8)
(hsep : 6*p*ρ^20+2*(n:ℝ)^(-(1:ℝ)/100)+ρ^20+3*d/2 ≤ 2*ρ)
(hd' : d/2 ≤ ρ^20) (heta : 2*(n:ℝ)^(-(1:ℝ)/100) < 2*ρ^20)
(hq : 2*t+6*p*ρ^20+2*(n:ℝ)^(-(1:ℝ)/100)+2*ρ^20 < q)
(hr : ∀ i < p, r i ∈ Icc t (2*t))
(hspacing : ∀ i j, i < j → j < p → 3*ρ ≤ r j-r i)
(hmass : ∀ i < p, (μ : Measure ℝ).real (Icc (r i-2*ρ^8) (r i+2*ρ^8)) ≤ ρ^4)
(hjump : ∀ z k, k < m → |overlap z (x (k+1))-overlap z (x k)| ≤ d)
(hcoverage : ∀ j < p, ∀ v : ℕ → Config n, (∀ i < j, v i ∈ pool i) →
∀ u ≤ m, (∀ i, j/B*B ≤ i → i < j → |overlap (v i) (x u)| ≤ 3*t) →
∃ w ∈ pool j, q ≤ overlap w (x u) ∧
∀ i, j/B*B ≤ i → i < j → |overlap (v i) w| ≤ 3*t)
(hsigned : ∀ i j, i < j → j < p → ∀ v ∈ pool i, ∀ w ∈ pool j, ∀ k ≤ m,
(![v,w,x k] : ReplicaConfig n 3) ∉ signedLockingSet n q b)
(hbank : ∀ i k j, i < k → k < j → j < p → ∀ v ∈ pool i, ∀ z ∈ pool k, ∀ w ∈ pool j,
(![z,v,w] : ReplicaConfig n 3) ∉ signedLockingSet n b b' ∧
(![z,v,flip w] : ReplicaConfig n 3) ∉ signedLockingSet n b b')
(hnarrow : ∀ i j, i < j → j < p → ∀ v ∈ pool i, ∀ w ∈ pool j, ∀ k ≤ m,
(![v,x k,w] : ReplicaConfig n 3) ∉ narrowLockingSet n μ t ρ)
(hpass : overlap (bankPick pool fallback B x t q 0 base 0) (x m) ≤ 0) :
∃ k ≤ m, ∀ a, a*B < p →
|overlap (greedyReference base (bankPick pool fallback B x t q) x r b m (a*B)) (x k)-r (a*B)| ≤ 6*p*ρ^20 := by
classical
let pick := bankPick pool fallback B x t q
let S := greedyRun base pick x r b m
have hε : 0 < ρ^20 := pow_pos hρ _
have hη : 0 ≤ 2*(n:ℝ)^(-(1:ℝ)/100) := by positivity
have htwo : 2*t < q := by
have hEp : 0 ≤ 6*(p:ℝ)*ρ^20 := by positivity
linarith only [hq,hEp,hε,hη]
have hmem (j i : ℕ) (hi : i < j) : (S j).refs i ∈ pool i := by
rw [show (S j).refs i=greedyReference base pick x r b m i from greedyRun_refs _ _ _ _ _ _ _ _ hi]
exact bankPick_mem pool fallback hf B x t q i _ _
have hfirst := bankPick_spec pool fallback B x t q 0 base 0
(hcoverage 0 hp base (by intros; omega) 0 (by omega) (by intros; omega))
have hanchor (j : ℕ) (hj : 0 < j) : q ≤ overlap ((S j).refs 0) (x 0) := by
rw [show (S j).refs 0=greedyReference base pick x r b m 0 from greedyRun_refs _ _ _ _ _ _ _ _ hj]
exact hfirst.1
have HI : ∀ j ≤ p, (S j).time ≤ m ∧ BlockProtected B j (S j).kept ∧
(∀ i ∈ (S j).kept, |overlap ((S j).refs i) (x (S j).time)-r i| ≤ 6*j*ρ^20) ∧
(∀ i ∈ (S j).kept, ∀ k ∈ (S j).kept, i < k →
|overlap ((S j).refs i) ((S j).refs k)-r i| ≤ 6*j*ρ^20) := by
intro j hjp
induction j with
| zero => simp [S,greedyRun,BlockProtected]
| succ j ih =>
have hjp' : j < p := by omega
obtain ⟨hτ,hblock,hpath,hpair⟩ := ih (by omega)
have hEr : 6*(j:ℝ)*ρ^20 ≤ 6*p*ρ^20 := by gcongr
have hA := greedyRun_range base pick x r b m j
have hcand := hcoverage j hjp' (S j).refs (hmem j) (S j).time hτ (by
intro i hil hij
have hlo : (j-1)/B*B ≤ j/B*B := Nat.mul_le_mul_right B (Nat.div_le_div_right (Nat.sub_le j 1))
have himem := hblock.1 i (hlo.trans hil) hij
have hh := hpath i himem
have hri := hr i (hij.trans hjp')
have hh' := abs_add_le (overlap ((S j).refs i) (x (S j).time)-r i) (r i)
rw [sub_add_cancel,abs_of_pos (ht.trans_le hri.1)] at hh'
linarith only [hh',hh,hri.2,hEr,hEst])
have hsel := bankPick_spec pool fallback B x t q j (S j).refs (S j).time hcand
let w := pick j (S j).refs (S j).time
have hw : w ∈ pool j := bankPick_mem pool fallback hf B x t q j _ _
by_cases hj0 : j=0
· subst j
have hband := firstCrossing_band (fun k => overlap w (x k)) (r 0) d m
(by change r 0 < overlap (bankPick pool fallback B x t q 0 base 0) (x 0); linarith only [hfirst.1,(hr 0 hp).2,htwo])
(by change overlap (bankPick pool fallback B x t q 0 base 0) (x m) ≤ r 0; linarith only [hpass,(hr 0 hp).1,ht])
(hjump w)
dsimp [w, S, greedyRun] at hband
change _ ≤ m ∧ _ ∧ _ ∧ _
simp only [S,greedyRun,zero_add,Nat.cast_one,mul_one]
simp only [Finset.empty_sdiff,Finset.insert_empty,Finset.mem_singleton,ite_true]
refine ⟨hband.1,?_,?_,?_⟩
· constructor
· intro i _ hi
have : i=0 := by omega
simpa only [Finset.mem_singleton] using this
· intro k hk
have : k*B=0 := by omega
simpa only [Finset.mem_singleton] using this
· intro i hi
subst i
simp only [Function.update_self]
rw [abs_le]
constructor <;> linarith only [hband.2.1,hband.2.2,hd',hε]
· intro i hi k hk hik
omega
· have hj : 0 < j := Nat.pos_of_ne_zero hj0
have Hstep := greedy_locking_step hn μ (S j).refs w x (S j).kept r B H j (S j).time
t ρ (6*j*ρ^20) d b b' q hj hB hHB hH ht hρ (by positivity) hd hb hpack hcsmall
(hEr.trans_lt hEst) (by linarith only [hsmall',hEr]) (by linarith only [hsep',hEr]) hwidth
(by linarith only [hsmall,hEr]) (by linarith only [hsep,hEr]) hd' heta
(by linarith only [hq,hEr]) (fun i hi => hr i (hi.trans_lt hjp'))
(fun i k hik hk => hspacing i k hik (hk.trans_lt hjp')) (fun i hi => hmass i (hi.trans hjp'))
hA hblock hpath hpair (hanchor j hj) hsel.1
(by
intro hmod
apply hsel.2 (j-1) _ (by omega)
have hh : j%B+j/B*B=j := by simpa only [Nat.mul_comm B] using Nat.mod_add_div j B
have hpos : 0 < j%B := Nat.pos_of_ne_zero hmod
omega)
(fun z k hk => hjump z k (hk.trans_le hτ))
(fun i hi => hsigned i j (hA i hi) hjp' _ (hmem j i (hA i hi)) w hw _ hτ)
(fun i hi k hk hik => hbank i k j hik (hA k hk) hjp' _ (hmem j i (hA i hi)) _ (hmem j k (hA k hk)) w hw)
(fun i hi k hk => hnarrow i j (hA i hi) hjp' _ (hmem j i (hA i hi)) w hw k (hk.trans hτ))
obtain ⟨hs,hcard,hprot,hzero,htime,hband,href,hnewpath⟩ := Hstep
have hinc : 6*(j:ℝ)*ρ^20+2*(n:ℝ)^(-(1:ℝ)/100)+2*ρ^20 ≤ 6*(j+1:ℕ)*ρ^20 := by
simp only [Nat.cast_add,Nat.cast_one]
nlinarith only [heta,hε]
have hmono : 6*(j:ℝ)*ρ^20 ≤ 6*(j+1:ℕ)*ρ^20 := by
simp only [Nat.cast_add,Nat.cast_one]; nlinarith only [hε]
change _ ≤ m ∧ _ ∧ _ ∧ _
simp only [S,greedyRun,ite_eq_right hj0]
refine ⟨htime.trans hτ,hprot,?_,?_⟩
· intro i hi
rcases Finset.mem_insert.mp hi with he | hi
· subst i
rw [Function.update_self,abs_le]
have hc : 2*ρ^20 ≤ 6*(j+1:ℕ)*ρ^20 := by
have hjr : (0:ℝ) ≤ j := by positivity
simp only [Nat.cast_add,Nat.cast_one]; nlinarith only [mul_nonneg hjr hε.le,hε]
constructor <;> linarith only [hband.1,hband.2,hd',hc]
· have him : i ∈ (S j).kept := (Finset.mem_sdiff.mp hi).1
rw [Function.update_of_ne (ne_of_lt (hA i him))]
exact (hnewpath i hi).trans hinc
· intro i hi k hk hik
have hil : i < j := by
rcases Finset.mem_insert.mp hk with rfl | hk
· exact hik
· exact hik.trans (hA k (Finset.mem_sdiff.mp hk).1)
have him : i ∈ (S j).kept\(S j).kept.filter (fun i => b < |overlap ((S j).refs i) w|) := by
exact (Finset.mem_insert.mp hi).resolve_left (ne_of_lt hil)
rw [Function.update_of_ne (ne_of_lt hil)]
rcases Finset.mem_insert.mp hk with rfl | hk
· rw [Function.update_self]
exact (href i him).trans (by linarith only [hinc,hε])
· have hkl := hA k (Finset.mem_sdiff.mp hk).1
rw [Function.update_of_ne (ne_of_lt hkl)]
exact (hpair i (Finset.mem_sdiff.mp him).1 k (Finset.mem_sdiff.mp hk).1 hik).trans hmono
obtain ⟨hτ,hprotected,hpath,_⟩ := HI p le_rfl
refine ⟨(S p).time,hτ,?_⟩
intro a ha
have hmem := hprotected.2 a ha
rw [← greedyRun_refs base pick x r b m p (a*B) ha]
exact hpath (a*B) hmem
end SK.Analytic
end
end
end OAI
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