| import Mathlib |
| import OAI.Geometry.Cannon.ScalarLimits.Words |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| section |
| section |
| section |
| section |
| section |
| section |
| open Set Topology |
| open Set MeasureTheory |
| open scoped BigOperators |
| open scoped NNReal ENNReal |
| open Set Metric |
| open Set Filter Topology |
| open Set Filter Topology Metric |
| open Set Metric Topology |
| open Set unitInterval |
|
|
| namespace CannonSecondMagnification |
|
|
| open Filter Topology Metric CannonVariation |
|
|
| theorem normalized_comparison {X : Type*} [TopologicalSpace X] |
| (W : ℕ → X → ℝ) (p : ℕ → X) (A sig : ℕ → ℝ) |
| (hsig : ∀ i, sig i ≠ 0) |
| (hc : ∀ i j, p i ≠ p j → ∀ x, x ≠ p i → x ≠ p j → |
| NonconstantAt (W i) x → W j x = W j (p i)) : |
| ∀ i j, p i ≠ p j → ∀ x, x ≠ p i → x ≠ p j → |
| NonconstantAt (fun z => (W i z - A i) / sig i) x → |
| (W j x - A j) / sig j = (W j (p i) - A j) / sig j := by |
| intro i j hij x hxi hxj hx |
| rw [hc i j hij x hxi hxj ((nonconstantAt_normalize _ _ _ (hsig i) _).mp hx)] |
|
|
| theorem impossible_nonconstant_limit {X : Type*} [MetricSpace X] |
| (F : ℕ → X → ℝ) (p : ℕ → X) (b : X) (W : X → ℝ) |
| (hp : Tendsto p atTop (𝓝 b)) (hinj : Function.Injective p) |
| (hW : ContinuousOn W ({b}ᶜ : Set X)) |
| (hlim : ∀ K : Set X, IsCompact K → b ∉ K → TendstoUniformlyOn F W atTop K) |
| (hcomp : ∀ i j, p i ≠ p j → ∀ x, x ≠ p i → x ≠ p j → |
| NonconstantAt (F i) x → F j x = F j (p i)) |
| {v w : X} (q : Path v w) (hqb : b ∉ range q) (hvar : W v < W w) : False := by |
| let d := (W w - W v) / 5 |
| have hd : 0 < d := div_pos (sub_pos.mpr hvar) (by norm_num) |
| have hWr : ContinuousOn W (range q) := hW.mono (by |
| intro x hx hxb |
| exact hqb (hxb ▸ hx)) |
| obtain ⟨v₁,w₁,q₁,hq₁,hv₁,hw₁,hW₁⟩ := value_subpath W q hWr |
| (W v + d) (W v + 2*d) (by linarith) (by linarith) (by dsimp [d]; linarith) |
| obtain ⟨v₂,w₂,q₂,hq₂,hv₂,hw₂,hW₂⟩ := value_subpath W q hWr |
| (W v + 3*d) (W v + 4*d) (by linarith) (by linarith) (by dsimp [d]; linarith) |
| let K := range q₁ ∪ range q₂ |
| have hK : IsCompact K := (isCompact_range q₁.continuous).union (isCompact_range q₂.continuous) |
| have hbK : b ∉ K := by |
| rintro (hb | hb) |
| · exact hqb (hq₁ hb) |
| · exact hqb (hq₂ hb) |
| have havoid : ∀ᶠ n in atTop, p n ∉ K := hp.eventually (hK.isClosed.isOpen_compl.mem_nhds hbK) |
| have he : ∀ᶠ n in atTop, ∀ z ∈ K, |W z - F n z| < d/4 := by |
| simpa only [Real.dist_eq] using (Metric.tendstoUniformlyOn_iff.mp (hlim K hK hbK) (d/4) (by positivity)) |
| obtain ⟨N,hN⟩ := eventually_atTop.1 (havoid.and he) |
| let i := N |
| let j := N+1 |
| have hi := hN i le_rfl |
| have hj := hN j (Nat.le_succ N) |
| have hv₁K : v₁ ∈ K := Or.inl (q₁.source_mem_range) |
| have hw₁K : w₁ ∈ K := Or.inl (q₁.target_mem_range) |
| have hv₂K : v₂ ∈ K := Or.inr (q₂.source_mem_range) |
| have hw₂K : w₂ ∈ K := Or.inr (q₂.target_mem_range) |
| have hi₁ : F i v₁ < F i w₁ := by |
| have ha := (abs_lt.mp (hi.2 v₁ hv₁K)) |
| have hb := (abs_lt.mp (hi.2 w₁ hw₁K)) |
| rw [hv₁] at ha |
| rw [hw₁] at hb |
| linarith |
| have hi₂ : F i v₂ < F i w₂ := by |
| have ha := (abs_lt.mp (hi.2 v₂ hv₂K)) |
| have hb := (abs_lt.mp (hi.2 w₂ hw₂K)) |
| rw [hv₂] at ha |
| rw [hw₂] at hb |
| linarith |
| obtain ⟨z₁,hz₁,hz₁var⟩ := path_meets_variation (F i) q₁ hi₁.ne |
| obtain ⟨z₂,hz₂,hz₂var⟩ := path_meets_variation (F i) q₂ hi₂.ne |
| have hz₁K : z₁ ∈ K := Or.inl hz₁ |
| have hz₂K : z₂ ∈ K := Or.inr hz₂ |
| have hpi : p i ≠ p j := fun h => (Nat.ne_add_one N) (hinj h) |
| have he₁ := hcomp i j hpi z₁ (fun h => hi.1 (h ▸ hz₁K)) (fun h => hj.1 (h ▸ hz₁K)) hz₁var |
| have he₂ := hcomp i j hpi z₂ (fun h => hi.1 (h ▸ hz₂K)) (fun h => hj.1 (h ▸ hz₂K)) hz₂var |
| have heq : F j z₁ = F j z₂ := he₁.trans he₂.symm |
| have hb₁ := (hW₁ z₁ hz₁).2 |
| have hb₂ := (hW₂ z₂ hz₂).1 |
| have ha := abs_lt.mp (hj.2 z₁ hz₁K) |
| have hb := abs_lt.mp (hj.2 z₂ hz₂K) |
| linarith |
|
|
| end CannonSecondMagnification |
|
|
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
