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a72760f | 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 | 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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