| import Mathlib.Tactic.Linarith |
| import Mathlib.Topology.Instances.Real.Lemmas |
|
|
| namespace OAI |
|
|
| namespace PiExponent |
|
|
| open Filter Topology |
|
|
| theorem determinant_bounds_inconsistent |
| (nu theta x b ear ean err collision d : ℝ) |
| (hnu : 1 < nu) (hb0 : 0 ≤ b) (hb : b ≤ theta) |
| (hgap : ear + ean + err < nu * (x - theta) - (1 - theta)) |
| (hcollision : 1 + ear + ean + err < collision) |
| (hlower : -(1 - b) - ear ≤ d) |
| (hupper : d ≤ ean + err + max (-collision) (-nu * (x - b))) : False := by |
| have hfirst : ean + err - collision < -(1 - b) - ear := by linarith |
| have hmono : |
| nu * (x - theta) - (1 - theta) ≤ nu * (x - b) - (1 - b) := by |
| nlinarith |
| have hsecond : ean + err - nu * (x - b) < -(1 - b) - ear := by linarith |
| rcases le_total (-collision) (-nu * (x - b)) with h | h |
| · rw [max_eq_right h] at hupper |
| linarith |
| · rw [max_eq_left h] at hupper |
| linarith |
|
|
| theorem asymptotic_determinant_bounds_inconsistent |
| (nu theta x ear ean collisionLimit : ℝ) |
| (b d err collision : ℕ → ℝ) |
| (hnu : 1 < nu) |
| (hgap : ear + ean < nu * (x - theta) - (1 - theta)) |
| (hcollision : 1 + ear + ean < collisionLimit) |
| (herr : Tendsto err atTop (𝓝 0)) |
| (hcol : Tendsto collision atTop (𝓝 collisionLimit)) |
| (hb0 : ∀ᶠ n in atTop, 0 ≤ b n) |
| (hb : ∀ᶠ n in atTop, b n ≤ theta) |
| (hlower : ∀ᶠ n in atTop, -(1 - b n) - ear ≤ d n) |
| (hupper : ∀ᶠ n in atTop, |
| d n ≤ ean + err n + max (-collision n) (-nu * (x - b n))) : False := by |
| have hsum : Tendsto (fun n => ear + ean + err n) atTop (𝓝 (ear + ean + 0)) := |
| tendsto_const_nhds.add herr |
| have hsmall : ∀ᶠ n in atTop, |
| ear + ean + err n < nu * (x - theta) - (1 - theta) := |
| hsum.eventually (Iio_mem_nhds (by simpa using hgap)) |
| have hdiff : Tendsto (fun n => collision n - (1 + ear + ean + err n)) |
| atTop (𝓝 (collisionLimit - (1 + ear + ean + 0))) := |
| hcol.sub (tendsto_const_nhds.add herr) |
| have hlarge : ∀ᶠ n in atTop, 0 < collision n - (1 + ear + ean + err n) := |
| hdiff.eventually (Ioi_mem_nhds (by linarith)) |
| have hfalse : ∀ᶠ n : ℕ in atTop, False := by |
| filter_upwards [hsmall, hlarge, hb0, hb, hlower, hupper] with n hs hc h0 ht hl hu |
| exact determinant_bounds_inconsistent nu theta x (b n) ear ean (err n) |
| (collision n) (d n) hnu h0 ht hs (by linarith) hl hu |
| exact hfalse.exists.elim fun _ h => h |
|
|
| end PiExponent |
|
|
| end OAI |
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|
