| import OAI.Geometry.CylinderCovering.Rational.InversionDefinitions |
| import OAI.Geometry.CylinderCovering.Convexity.ThresholdInversion |
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| namespace OAI |
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|
| open Set |
| open scoped NNReal |
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| namespace SlopeRational |
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| lemma inverseRectangle_eq_box (d : ℝ) : |
| inverseRectangle d = |
| {p : Plane | ∀ i : Fin 2, p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)} := by |
| ext p |
| simp [inverseRectangle, Fin.forall_fin_two, abs_le, and_assoc] |
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| lemma inverseRectangle_isClosed (d : ℝ) : IsClosed (inverseRectangle d) := by |
| unfold inverseRectangle |
| have h0 : Continuous (fun p : Plane => |p 0|) := by fun_prop |
| have h1 : Continuous (fun p : Plane => p 1) := by fun_prop |
| exact (isClosed_le h0 continuous_const).inter |
| ((isClosed_le continuous_const h1).inter (isClosed_le h1 continuous_const)) |
|
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| lemma inverseRectangle_isCompact {d : ℝ} (hd : 0 < d) : |
| IsCompact (inverseRectangle d) := by |
| apply Metric.isCompact_iff_isClosed_bounded.mpr |
| refine ⟨inverseRectangle_isClosed d, isBounded_iff_forall_norm_le.mpr ?_⟩ |
| refine ⟨2 + d, ?_⟩ |
| intro p hp |
| rcases hp with ⟨hp0, hp1lo, hp1hi⟩ |
| have hp1 : |p 1| ≤ 1 + d := by |
| apply abs_le.mpr |
| constructor <;> linarith |
| have hnorm := EuclideanSpace.norm_sq_eq p |
| simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hnorm |
| have hp0sq : (p 0) ^ 2 ≤ 1 := by |
| have hs := (sq_le_sq₀ (abs_nonneg (p 0)) (by norm_num : (0 : ℝ) ≤ 1)).mpr hp0 |
| simpa only [sq_abs, one_pow] using hs |
| have hp1sq : (p 1) ^ 2 ≤ (1 + d) ^ 2 := by |
| have hs := (sq_le_sq₀ (abs_nonneg (p 1)) (by positivity : 0 ≤ 1 + d)).mpr hp1 |
| simpa only [sq_abs] using hs |
| nlinarith [norm_nonneg p] |
|
|
| lemma inverseRectangle_convex (d : ℝ) : Convex ℝ (inverseRectangle d) := by |
| rw [inverseRectangle_eq_box] |
| have heq : {p : Plane | ∀ i : Fin 2, p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)} = |
| ⋂ i : Fin 2, {p : Plane | p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)} := by |
| ext p |
| simp |
| rw [heq] |
| exact convex_iInter fun i => |
| (convex_Icc _ _).linear_preimage (PiLp.projₗ 2 (fun _ : Fin 2 => ℝ) i) |
|
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| lemma inverseRectangle_denominator_pos {d : ℝ} (hd : 0 < d) |
| {p : Plane} (hp : p ∈ inverseRectangle d) : 0 < radius (p 1) + d := by |
| have ht := hp.2.2 |
| unfold radius |
| linarith |
|
|
| lemma field_contDiffOn_inverseRectangle {d : ℝ} (hd : 0 < d) : |
| ContDiffOn ℝ 1 (field d) (inverseRectangle d) := by |
| intro p hp |
| apply ContDiffAt.contDiffWithinAt |
| have hden : 0 < 1 / 2 - p 1 + d := inverseRectangle_denominator_pos hd hp |
| unfold field vec2 |
| apply PiLp.contDiff_toLp.contDiffAt.comp p |
| apply contDiffAt_pi.mpr |
| intro i |
| fin_cases i |
| · change ContDiffAt ℝ 1 (fun z : Plane => a d (z 0) (z 1)) p |
| unfold a radius |
| fun_prop (disch := positivity) |
| · change ContDiffAt ℝ 1 (fun z : Plane => b d (z 0) (z 1)) p |
| unfold b radius |
| fun_prop (disch := positivity) |
|
|
| lemma rectangleInversion_of_contDiff {d : ℝ} (hd : 0 < d) |
| (hregular : ContDiffOn ℝ 1 (field d) (inverseRectangle d)) : RectangleInversionAt d := by |
| obtain ⟨L, hL⟩ := hregular.exists_lipschitzOnWith (by norm_num) |
| (inverseRectangle_convex d) (inverseRectangle_isCompact hd) |
| obtain ⟨B₀, hB₀⟩ := (inverseRectangle_isCompact hd).exists_bound_of_continuousOn |
| hregular.continuousOn |
| let B := max B₀ 0 |
| have hB : 0 ≤ B := le_max_right _ _ |
| have hbound : ∀ p ∈ inverseRectangle d, ‖field d p‖ ≤ B := by |
| intro p hp |
| exact (hB₀ p hp).trans (le_max_left _ _) |
| let margin := min (1 / 2 : ℝ) (d / 2) |
| have hm : 0 < margin := lt_min (by norm_num) (by positivity) |
| obtain ⟨δ, hδ, hsmall⟩ := CylinderCovering.ThresholdInversion.exists_small_tilt |
| (L : ℝ) B margin L.coe_nonneg hB hm |
| refine ⟨δ, hδ, ?_⟩ |
| intro s hs |
| obtain ⟨hsL, hsB⟩ := hsmall s hs |
| constructor |
| · intro y t hy ht |
| have hbox := CylinderCovering.ThresholdInversion.existsUnique_inverse_on_box_smul |
| (![-1, -1] : Fin 2 → ℝ) (![1, 1 / 2 + d / 2] : Fin 2 → ℝ) hm.le |
| (W := field d) (L := (L : ℝ)) (B := B) (t := s) L.coe_nonneg hB |
| have hlip : ∀ p : Plane, (∀ i, p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)) → |
| ∀ q : Plane, (∀ i, q i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)) → |
| ‖field d p - field d q‖ ≤ (L : ℝ) * ‖p - q‖ := by |
| intro p hp q hq |
| have hp' : p ∈ inverseRectangle d := by rw [inverseRectangle_eq_box]; exact hp |
| have hq' : q ∈ inverseRectangle d := by rw [inverseRectangle_eq_box]; exact hq |
| simpa only [dist_eq_norm] using hL.dist_le_mul p hp' q hq' |
| have hb : ∀ p : Plane, (∀ i, p i ∈ Icc (![-1, -1] i) (![1, 1 / 2 + d / 2] i)) → |
| ‖field d p‖ ≤ B := by |
| intro p hp |
| exact hbound p (by rwa [inverseRectangle_eq_box]) |
| have htarget : ∀ i : Fin 2, |
| ![-1, -1] i + margin ≤ vec2 y t i ∧ |
| vec2 y t i ≤ ![1, 1 / 2 + d / 2] i - margin := by |
| have hm1 : margin ≤ 1 / 2 := min_le_left _ _ |
| have hmd : margin ≤ d / 2 := min_le_right _ _ |
| have hy' := abs_le.mp hy |
| have ht' := abs_le.mp ht |
| intro i |
| fin_cases i |
| · change -1 + margin ≤ y ∧ y ≤ 1 - margin |
| constructor <;> linarith |
| · change -1 + margin ≤ t ∧ t ≤ 1 / 2 + d / 2 - margin |
| constructor <;> linarith |
| have hex := hbox hlip hb (by linarith) hsB (vec2 y t) htarget |
| simpa only [inverseRectangle_eq_box, tiltedMap, Set.mem_ofPred_eq] using hex |
| · intro p hp q hq hhorizontal hvertical |
| apply CylinderCovering.ThresholdInversion.inverse_coordinate_strict (i := (1 : Fin 2)) |
| (c := |s| * (L : ℝ)) (mul_nonneg (abs_nonneg _) L.coe_nonneg) hsL |
| (g := fun z => s • field d z) (y := tiltedMap d s p) (z := tiltedMap d s q) |
| rfl rfl |
| · rw [← smul_sub, norm_smul, Real.norm_eq_abs] |
| have hLip := hL.dist_le_mul q hq p hp |
| simp only [dist_eq_norm] at hLip |
| calc |
| |s| * ‖field d q - field d p‖ ≤ |s| * ((L : ℝ) * ‖q - p‖) := |
| mul_le_mul_of_nonneg_left hLip (abs_nonneg _) |
| _ = |s| * (L : ℝ) * ‖q - p‖ := by ring |
| · intro j hj |
| fin_cases j |
| · exact hhorizontal.symm |
| · exact (hj rfl).elim |
| · exact hvertical |
|
|
| theorem rectangleInversion_main : RectangleInversionStatement := by |
| intro d hd |
| exact rectangleInversion_of_contDiff hd (field_contDiffOn_inverseRectangle hd) |
|
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| end SlopeRational |
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| end OAI |
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