| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.BackwardBarrier |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def lastCrossing (f : ℕ → ℝ) (r : ℝ) (u : ℕ) : ℕ := |
| Nat.findGreatest (fun k => f k ≤ r) u |
|
|
| theorem lastCrossing_le (f : ℕ → ℝ) (r : ℝ) (u : ℕ) : lastCrossing f r u ≤ u := |
| Nat.findGreatest_le u |
|
|
| theorem lastCrossing_spec (f : ℕ → ℝ) (r : ℝ) (u : ℕ) |
| (hex : ∃ k ≤ u,f k ≤ r) : f (lastCrossing f r u) ≤ r := by |
| obtain ⟨k,hk,h⟩ := hex |
| exact Nat.findGreatest_spec (P:=fun k => f k ≤ r) hk h |
|
|
| theorem lastCrossing_after (f : ℕ → ℝ) (r : ℝ) (u k : ℕ) |
| (hk : lastCrossing f r u < k) (hku : k ≤ u) : r < f k := by |
| exact lt_of_not_ge ((Nat.findGreatest_eq_iff.mp (rfl : Nat.findGreatest (fun k => f k ≤ r) u = _)).2.2 hk hku) |
|
|
| theorem lastCrossing_band (f : ℕ → ℝ) (r d : ℝ) (u : ℕ) |
| (hend : r < f u) (hex : ∃ k ≤ u,f k ≤ r) |
| (hjump : ∀ k, k < u → |f (k+1)-f k| ≤ d) : |
| f (lastCrossing f r u) ∈ Icc (r-d) r := by |
| have hl := lastCrossing_le f r u |
| have hs := lastCrossing_spec f r u hex |
| have hlt : lastCrossing f r u < u := lt_of_le_of_ne hl (by intro he; rw [he] at hs; linarith) |
| have ha := lastCrossing_after f r u (lastCrossing f r u+1) (by omega) (by omega) |
| have hj := hjump _ hlt |
| exact ⟨by linarith only [ha,(abs_le.mp hj).2],hs⟩ |
|
|
| theorem lastCrossing_search (f : ℕ → ℝ) (r d : ℝ) (u : ℕ) |
| (hd : 0 ≤ d) (hend : r < f u) |
| (hjump : ∀ k, k < u → |f (k+1)-f k| ≤ d) : |
| ∀ k,lastCrossing f r u ≤ k → k ≤ u → r-d ≤ f k := by |
| intro k hk hku |
| by_cases hex : ∃ i ≤ u,f i ≤ r |
| · by_cases he : lastCrossing f r u=k |
| · rw [← he]; exact (lastCrossing_band f r d u hend hex hjump).1 |
| · exact (by linarith only [lastCrossing_after f r u k (lt_of_le_of_ne hk he) hku,hd]) |
| · have h : r < f k := lt_of_not_ge (by intro h; exact hex ⟨k,hku,h⟩) |
| linarith only [h,hd] |
|
|
| theorem lastCrossing_zero_of_none (f : ℕ → ℝ) (r : ℝ) (u : ℕ) |
| (h : ¬∃ k ≤ u,f k ≤ r) : lastCrossing f r u=0 := by |
| apply Nat.findGreatest_eq_zero_iff.mpr |
| intro k _ hk hkr |
| exact h ⟨k,hk,hkr⟩ |
|
|
| theorem anchored_backwards_crossing {n : ℕ} (μ : ProbabilityMeasure ℝ) |
| (v w : Config n) (x : ℕ → Config n) (t₀ ρ E d rᵢ rⱼ q : ℝ) (u : ℕ) |
| (ht : 0 < t₀) (hρ : 0 < ρ) (hri : rᵢ ∈ Icc t₀ (2*t₀)) |
| (hspacing : 3*ρ ≤ rⱼ-rᵢ) (hwidth : ρ^8 < t₀/2) |
| (hsmall : E+ρ^20+d/2 < ρ^8) (hsep : E+ρ^20+3*d/2 ≤ 2*ρ) |
| (hd : 0 ≤ d) (hd' : d/2 ≤ ρ^20) (hq : rᵢ+E+2*ρ^20 < q) |
| (hmass : (μ : Measure ℝ).real (Icc (rᵢ-2*ρ^8) (rᵢ+2*ρ^8)) ≤ ρ^4) |
| (href : |overlap v w-rᵢ| ≤ E) |
| (hanchor : q ≤ overlap v (x 0)) (_htarget : rⱼ < overlap w (x u)) |
| (hend : |overlap v (x u)-overlap v w| < 2*ρ^20) |
| (hjump : ∀ z k, k < u → |overlap z (x (k+1))-overlap z (x k)| ≤ d) |
| (havoid : ∀ k, k ≤ u → (![v,x k,w] : ReplicaConfig n 3)∉narrowLockingSet n μ t₀ ρ) : |
| ∃ k ≤ u,overlap w (x k) ≤ rⱼ := by |
| by_contra H |
| have hs : ∀ k,0 ≤ k → k ≤ u → rⱼ-d ≤ overlap w (x k) := by |
| intro k _ hk |
| have hh : rⱼ < overlap w (x k) := lt_of_not_ge (fun hh => H ⟨k,hk,hh⟩) |
| linarith only [hh,hd] |
| have HH := narrow_backwards_transfer μ v w x t₀ ρ E d rᵢ rⱼ (Nat.zero_le u) |
| ht hρ hri hspacing hwidth hsmall hsep hd' hmass href hend |
| (fun k _ hk => hjump v k hk) hs (fun k _ hk => havoid k hk) 0 (by omega) (by omega) |
| linarith only [(abs_lt.mp HH).2,(abs_le.mp href).2,hanchor,hq] |
|
|
| def firstCrossing (f : ℕ → ℝ) (r : ℝ) (m : ℕ) : ℕ := |
| if h : ∃ k, k ≤ m ∧ f k ≤ r then Nat.find h else 0 |
|
|
| theorem firstCrossing_band (f : ℕ → ℝ) (r d : ℝ) (m : ℕ) |
| (hstart : r < f 0) (hend : f m ≤ r) |
| (hjump : ∀ k, k < m → |f (k+1)-f k| ≤ d) : |
| firstCrossing f r m ≤ m ∧ f (firstCrossing f r m) ∈ Icc (r-d) r := by |
| have hex : ∃ k, k ≤ m ∧ f k ≤ r := ⟨m,le_rfl,hend⟩ |
| rw [firstCrossing,dite_eq_left hex] |
| have hs := Nat.find_spec hex |
| have hz : 0 < Nat.find hex := by |
| by_contra H |
| have he : Nat.find hex=0 := by omega |
| rw [he] at hs |
| linarith only [hs.2,hstart] |
| have hp : r < f (Nat.find hex-1) := by |
| apply lt_of_not_ge |
| intro h |
| exact Nat.find_min hex (by omega : Nat.find hex-1 < Nat.find hex) ⟨by omega,h⟩ |
| have hj := hjump (Nat.find hex-1) (by omega) |
| rw [Nat.sub_add_cancel hz] at hj |
| exact ⟨hs.1,by linarith only [hp,(abs_le.mp hj).1],hs.2⟩ |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
