| import OAI.Geometry.CylinderCovering.Angular.CartesianDirect |
| import OAI.Geometry.CylinderCovering.Angular.CartesianObstruction |
|
|
| namespace OAI |
|
|
| namespace AngularCartesianDirect |
| open CylinderCovering.CartesianCaps |
|
|
| theorem selected_pair_cover {η M : ℝ} (hη : 0 < η) (hM : 0 ≤ M) : |
| ∃ τ₀ : ℝ, 0 < τ₀ ∧ ∀ τ : ℝ, 0 < τ → τ ≤ τ₀ → |
| ∀ x y s l r l' r' S T : ℝ, |
| |x| ≤ 1 → |y| ≤ 1 → 0 ≤ s → s ≤ 1 → |
| -1 ≤ l → l ≤ r → r ≤ 1 → -1 ≤ l' → l' ≤ r' → r' ≤ 1 → |
| r-l ≤ M*τ^2 → r'-l' ≤ M*τ^2 → |
| 0 ≤ S → 0 ≤ T → S = s-τ*x*((l+r)/4) → T = 1-s+τ*y*((l'+r')/4) → |
| l*S ≤ y-τ*x*((1+l*r)/4) → y-τ*x*((1+l*r)/4) ≤ r*S → |
| l'*T ≤ x+τ*y*((1+l'*r')/4) → x+τ*y*((1+l'*r')/4) ≤ r'*T → |
| (y-τ*x*((1+l*r)/4), S) ∈ domain τ η 1 ∨ |
| (x+τ*y*((1+l'*r')/4), T) ∈ domain τ η 1 := by |
| obtain ⟨a, ha, hsmall⟩ := smallness_radius hη hM |
| refine ⟨min a (η/240), lt_min ha (by positivity), ?_⟩ |
| intro τ hτ hτle x y s l r l' r' S T hx hy hs0 hs1 hl hlr hr hl' hlr' hr' |
| hmesh hmesh' hS0 hT0 hS hT hleft hright hleft' hright' |
| obtain ⟨hτ1, hmarg, houter⟩ := hsmall τ hτ.le (hτle.trans (min_le_left _ _)) |
| have hfinal : τ ≤ η/240 := hτle.trans (min_le_right _ _) |
| by_contra! hfail |
| have hδ : |τ*x| ≤ τ := by |
| rw [abs_mul, abs_of_pos hτ] |
| simpa only [mul_one] using mul_le_mul_of_nonneg_left hx hτ.le |
| have hδ' : |-(τ*y)| ≤ τ := by |
| rw [abs_neg, abs_mul, abs_of_pos hτ] |
| simpa only [mul_one] using mul_le_mul_of_nonneg_left hy hτ.le |
| have hgain := selected_domain_gain hl hlr hr hy hδ hτ.le hτ1 hM hmesh hmarg |
| houter hS0 hS hleft hright hfail.1 |
| have hT' : T = (1-s)-(-(τ*y))*((l'+r')/4) := by rw [hT]; ring |
| have hleft'' : l'*T ≤ x-(-(τ*y))*((1+l'*r')/4) := by |
| simpa only [neg_mul, sub_neg_eq_add] using hleft' |
| have hright'' : x-(-(τ*y))*((1+l'*r')/4) ≤ r'*T := by |
| simpa only [neg_mul, sub_neg_eq_add] using hright' |
| have hfail' : (x-(-(τ*y))*((1+l'*r')/4), T) ∉ domain τ η 1 := by |
| simpa only [neg_mul, sub_neg_eq_add] using hfail.2 |
| have hgain' := selected_domain_gain hl' hlr' hr' hx hδ' hτ.le hτ1 hM hmesh' |
| hmarg houter hT0 hT' hleft'' hright'' hfail' |
| have hQ : s^2-(τ*x)*y+(τ*x)^2/4 = s^2-τ*x*y+τ^2*x^2/4 := by ring |
| have hQ' : (1-s)^2-(-(τ*y))*x+(-(τ*y))^2/4 = |
| (1-s)^2+τ*x*y+τ^2*y^2/4 := by ring |
| rw [hQ] at hgain |
| rw [hQ'] at hgain' |
| rcases discriminant_alternative hτ hτ1 hfinal hx hy hs0 hs1 with hfirst | hsecond |
| · exact (not_lt_of_ge hfirst) hgain |
| · exact (not_lt_of_ge hsecond) hgain' |
| |
| end AngularCartesianDirect |
| |
| end OAI |
| |
