| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.Crossing |
| import OAI.Probability.SKBarriers.Dynamics.ConflictSuffix |
| import OAI.Probability.SKBarriers.Dynamics.ProtectedStep |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem greedy_locking_step {n : ℕ} (hn : 0 < n) (μ : ProbabilityMeasure ℝ) |
| (v : ℕ → Config n) (w : Config n) (x : ℕ → Config n) (A : Finset ℕ) |
| (r : ℕ → ℝ) (B H j u : ℕ) (t ρ E d b b' q : ℝ) |
| (hj : 0 < j) (hB : 0 < B) (hHB : H < B) (hH : ⌈2/b^2⌉₊ ≤ H) |
| (ht : 0 < t) (hρ : 0 < ρ) (hE : 0 ≤ E) (hd : 0 ≤ d) (hb : 0 < b) |
| (hpack : 3*t < b^2/2) (hcsmall : 3*t < b) (hEst : E < t) |
| (hsmall' : 2*t+E+2*(n:ℝ)^(-(1:ℝ)/100) < b') |
| (hsep' : 2*E+4*(n:ℝ)^(-(1:ℝ)/100) < 3*ρ) |
| (hwidth : ρ^8 < t/2) |
| (hsmall : E+2*(n:ℝ)^(-(1:ℝ)/100)+ρ^20+d/2 < ρ^8) |
| (hsep : E+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+E+2*(n:ℝ)^(-(1:ℝ)/100)+2*ρ^20 < q) |
| (hr : ∀ i ≤ j, r i ∈ Icc t (2*t)) |
| (hspacing : ∀ i k, i < k → k ≤ j → 3*ρ ≤ r k-r i) |
| (hmass : ∀ i < j, (μ : Measure ℝ).real (Icc (r i-2*ρ^8) (r i+2*ρ^8)) ≤ ρ^4) |
| (hA : ∀ i ∈ A, i < j) (hblock : BlockProtected B j A) |
| (hpath : ∀ i ∈ A, |overlap (v i) (x u)-r i| ≤ E) |
| (hpair : ∀ i ∈ A, ∀ k ∈ A, i < k → |overlap (v i) (v k)-r i| ≤ E) |
| (hanchor : q ≤ overlap (v 0) (x 0)) (htarget : q ≤ overlap w (x u)) |
| (hcell : j%B ≠ 0 → |overlap (v (j-1)) w| ≤ 3*t) |
| (hjump : ∀ z k, k < u → |overlap z (x (k+1))-overlap z (x k)| ≤ d) |
| (hsigned : ∀ i ∈ A, (![v i,w,x u] : ReplicaConfig n 3) ∉ signedLockingSet n q b) |
| (hbank : ∀ i ∈ A, ∀ k ∈ A, i < k → |
| (![v k,v i,w] : ReplicaConfig n 3) ∉ signedLockingSet n b b' ∧ |
| (![v k,v i,flip w] : ReplicaConfig n 3) ∉ signedLockingSet n b b') |
| (hnarrow : ∀ i ∈ A, ∀ k ≤ u, |
| (![v i,x k,w] : ReplicaConfig n 3) ∉ narrowLockingSet n μ t ρ) : |
| let D := A.filter (fun i => b < |overlap (v i) w|) |
| let C := A\D |
| let τ := lastCrossing (fun k => overlap w (x k)) (r j) u |
| IsRetainedSuffix A D ∧ D.card ≤ H ∧ BlockProtected B (j+1) (insert j C) ∧ |
| 0 ∈ C ∧ τ ≤ u ∧ overlap w (x τ) ∈ Icc (r j-d) (r j) ∧ |
| (∀ i ∈ C, |overlap (v i) w-r i| ≤ E+2*(n:ℝ)^(-(1:ℝ)/100)) ∧ |
| (∀ i ∈ C, |overlap (v i) (x τ)-r i| ≤ E+2*(n:ℝ)^(-(1:ℝ)/100)+2*ρ^20) := by |
| classical |
| let F := 2*(n:ℝ)^(-(1:ℝ)/100) |
| have hF : 0 ≤ F := by dsimp [F]; positivity |
| let D := A.filter (fun i => b < |overlap (v i) w|) |
| let C := A\D |
| have hc (i : ℕ) (hi : i ∈ C) : i ∈ A ∧ |overlap (v i) w| ≤ b := by |
| have H := Finset.mem_sdiff.mp hi |
| exact ⟨H.1,le_of_not_gt (fun hh => H.2 (Finset.mem_filter.mpr ⟨H.1,hh⟩))⟩ |
| have htransfer (i : ℕ) (hi : i ∈ A) (hcomp : |overlap (v i) w| ≤ b) : |
| |overlap (v i) w-overlap (v i) (x u)| ≤ F := by |
| apply signed_locking_of_not_mem (v i) w (x u) q b htarget hcomp _ (hsigned i hi) |
| have hh := abs_add_le (overlap (v i) (x u)-r i) (r i) |
| have hri := hr i (hA i hi).le |
| rw [sub_add_cancel,abs_of_pos (ht.trans_le hri.1)] at hh |
| linarith only [hh,hpath i hi,hri.2,hEst,hcsmall] |
| have href (i : ℕ) (hi : i ∈ A) (hcomp : |overlap (v i) w| ≤ b) : |
| |overlap (v i) w-r i| ≤ E+F := by |
| have hh := abs_add_le (overlap (v i) w-overlap (v i) (x u)) (overlap (v i) (x u)-r i) |
| rw [sub_add_sub_cancel] at hh |
| linarith only [hh,htransfer i hi hcomp,hpath i hi] |
| have hs : IsRetainedSuffix A D := by |
| apply conflicts_are_suffix A v w r b b' t E F ρ hE ht.le hsmall' (by dsimp [F]; linarith only [hsep']) |
| (fun i hi => hr i (hA i hi).le) (fun i hi k hk hik => hspacing i k hik (hA k hk).le) hpair href _ hF |
| intro i hi k hk hik hconf hki hkw |
| exact magnitude_locking_of_signed_avoid (v k) (v i) w b b' hconf.le hki hkw |
| (hbank i hi k hk hik).1 (hbank i hi k hk hik).2 |
| have hcard : D.card ≤ H := by |
| apply (conflict_filter_card hn A v w b (3*t) hb (by positivity) hpack _).trans hH |
| intro i hi k hk hik |
| have HH : |overlap (v i) (v k)-r (min i k)| ≤ E := by |
| rcases lt_or_gt_of_ne hik with h | h |
| · simpa only [min_eq_left h.le] using hpair i hi k hk h |
| · simpa only [min_eq_right h.le,overlap_comm (v k) (v i)] using hpair k hk i hi h |
| have hm : min i k ≤ j := (min_le_left _ _).trans (hA i hi).le |
| have hh := abs_add_le (overlap (v i) (v k)-r (min i k)) (r (min i k)) |
| rw [sub_add_cancel,abs_of_pos (ht.trans_le (hr _ hm).1)] at hh |
| linarith only [hh,HH,(hr _ hm).2,hEst] |
| have hp := protected_block_step B H j hB hHB A D hA hblock hs hcard (by |
| intro hmod hmem |
| have hh := (Finset.mem_filter.mp hmem).2 |
| linarith only [hh,hcell hmod,hcsmall]) |
| have hzero : 0 ∈ C := hp.2 hj |
| have hqj : r j < overlap w (x u) := by |
| have hpow : 0 ≤ ρ^20 := by positivity |
| dsimp [F] at hF |
| linarith only [(hr j le_rfl).2,htarget,hq,hE,hF,hpow] |
| have hsend (i : ℕ) (hi : i ∈ C) : |overlap (v i) (x u)-overlap (v i) w| < 2*ρ^20 := by |
| rw [abs_sub_comm] |
| exact (htransfer i (hc i hi).1 (hc i hi).2).trans_lt heta |
| have hex := anchored_backwards_crossing μ (v 0) w x t ρ (E+F) d (r 0) (r j) q u |
| ht hρ (hr 0 (by omega)) (hspacing 0 j hj le_rfl) hwidth hsmall hsep hd hd' |
| (by dsimp [F]; linarith only [(hr 0 (by omega)).2,hq]) (hmass 0 hj) |
| (href 0 (hc 0 hzero).1 (hc 0 hzero).2) hanchor hqj (hsend 0 hzero) hjump |
| (hnarrow 0 (hc 0 hzero).1) |
| have hlu := lastCrossing_le (fun k => overlap w (x k)) (r j) u |
| refine ⟨hs,hcard,hp.1,hzero,hlu,lastCrossing_band _ _ _ _ hqj hex (hjump w),?_,?_⟩ |
| · intro i hi |
| exact href i (hc i hi).1 (hc i hi).2 |
| · intro i hi |
| have hiA := (hc i hi).1 |
| have Hn := narrow_backwards_transfer μ (v i) w x t ρ (E+F) d (r i) (r j) hlu |
| ht hρ (hr i (hA i hiA).le) (hspacing i j (hA i hiA) le_rfl) hwidth hsmall hsep hd' |
| (hmass i (hA i hiA)) (href i hiA (hc i hi).2) (hsend i hi) |
| (fun k _ hk => hjump (v i) k hk) |
| (lastCrossing_search _ _ _ _ hd hqj (hjump w)) |
| (fun k _ hk => hnarrow i hiA k hk) _ le_rfl hlu |
| have HH := abs_add_le (overlap (v i) (x (lastCrossing (fun k => overlap w (x k)) (r j) u))-overlap (v i) w) |
| (overlap (v i) w-r i) |
| rw [sub_add_sub_cancel] at HH |
| have HH2 := href i hiA (hc i hi).2 |
| dsimp [F] at HH2 |
| linarith only [HH,Hn,HH2] |
|
|
| end SK.Analytic |
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| end |
| end |
|
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| end OAI |
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