File size: 4,207 Bytes
8e4d533 | 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 | 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
|