| import OAI.Geometry.CylinderCovering.Physical.ProjectionDictionary |
| import OAI.Geometry.CylinderCovering.Radial.PartitionAffine |
|
|
| namespace OAI |
|
|
| open Set |
| noncomputable |
| namespace |
| open |
|
|
| lemma ℝ ℝ ℝ → ℝ |
| (N ℕ |
| mapToCanonical |
| upperSweepOn |
| (RadialRiemann.cell |
| have |
| ext |
| constructor |
| · rintro ⟨ ⟨⟩⟩ |
| have /b)=s := by field_simp |
| refine ⟨s/ |
| 2*rho, ⟨by ⟩/b, ?_, ?_⟩ |
| · change 2*rho ≤ 2*T (b*(s/ |
| rw |
| linarith |
| · change || ≤ |
| norm_num |
| rw |
| norm_num |
| apply ₀ |
| linarith |
| · ext |
| fin_cases ;> simp [mapToCanonical, upperPoint, point, height, |
| effectiveTilt] <;> field_simp |
| ring |
| · rintro ⟨r, hr, rho, hrho, x, hx, hz⟩ |
| refine ⟨upperPoint b eta (rho/2) (b*r) (b*x/2), ?_, ?_⟩ |
| · refine ⟨b*r, ?_, (rho/2), ⟨by linarith [hrho.1], by linarith [hrho.2]⟩, (b*x/2), ?_, rfl⟩ |
| · apply (RadialRiemann.div_mem_symmetric_cell_iff hb N i (b*r)).1 |
| simpa [hb0] using hr |
| · norm_num only [axialBound, Bool.true_eq, ite_true, mul_one] at hx |
| change |(b*x/2)| ≤ b |
| rw [abs_div, abs_mul, abs_of_pos hb] |
| norm_num |
| nlinarith |
| · rw [hz] |
| ext j |
| fin_cases j <;> simp [mapToCanonical, upperPoint, point, height, |
| effectiveTilt] <;> field_simp |
| ring |
|
|
| lemma halfheight_lowerLocal_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (i : Fin N) : |
| mapToCanonical b '' lowerSweepOn false b eta T (RadialRiemann.cell (-b) b N i) = |
| lowerSweepOn true 1 (effectiveTilt b eta) (fun r => 2*T (b*r)) |
| (RadialRiemann.cell (-1) 1 N i) := by |
| have hb0 := ne_of_gt hb |
| ext z |
| constructor |
| · rintro ⟨v, ⟨s, hs, rho, hrho, x, hx, rfl⟩, rfl⟩ |
| have hbs : b*(s/b)=s := by field_simp |
| refine ⟨s/b, (RadialRiemann.div_mem_symmetric_cell_iff hb N i s).2 hs, |
| 2*rho, ⟨by linarith [hrho.1], ?_⟩, 2*x/b, ?_, ?_⟩ |
| · change 2*rho ≤ 2*T (b*(s/b)) |
| rw [hbs] |
| linarith [hrho.2] |
| · change |x| ≤ b at hx |
| norm_num only [axialBound, Bool.true_eq, ite_true, mul_one] |
| rw [abs_div, abs_mul, abs_of_pos hb] |
| norm_num |
| apply (div_le_iff₀ hb).mpr |
| linarith |
| · ext j |
| fin_cases j <;> simp [mapToCanonical, lowerPoint, point, height, |
| effectiveTilt] <;> field_simp |
| ring |
| · rintro ⟨r, hr, rho, hrho, x, hx, hz⟩ |
| refine ⟨lowerPoint false b eta (rho/2) (b*r) (b*x/2), ?_, ?_⟩ |
| · refine ⟨b*r, ?_, (rho/2), ⟨by linarith [hrho.1], by linarith [hrho.2]⟩, (b*x/2), ?_, rfl⟩ |
| · apply (RadialRiemann.div_mem_symmetric_cell_iff hb N i (b*r)).1 |
| simpa [hb0] using hr |
| · norm_num only [axialBound, Bool.true_eq, ite_true, mul_one] at hx |
| change |(b*x/2)| ≤ b |
| rw [abs_div, abs_mul, abs_of_pos hb] |
| norm_num |
| nlinarith |
| · rw [hz] |
| ext j |
| fin_cases j <;> simp [mapToCanonical, lowerPoint, point, height, |
| effectiveTilt] <;> field_simp |
| ring |
|
|
| lemma centered_upperLocal_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (i : Fin N) : |
| mapCenteredToCanonical b '' upperSweepOn true b eta T (RadialRiemann.cell (-b) b N i) = |
| upperSweepOn true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| (RadialRiemann.cell (-1) 1 N i) := by |
| have hb0 := ne_of_gt hb |
| ext z |
| constructor |
| · rintro ⟨v, ⟨s, hs, rho, hrho, x, hx, rfl⟩, rfl⟩ |
| have hbs : b*(s/b)=s := by field_simp |
| refine ⟨s/b, (RadialRiemann.div_mem_symmetric_cell_iff hb N i s).2 hs, |
| rho, ⟨hrho.1, ?_⟩, x/b, ?_, ?_⟩ |
| · change rho ≤ T (b*(s/b)) |
| rw [hbs] |
| linarith [hrho.2] |
| · change |x| ≤ 2*b at hx |
| norm_num only [axialBound, Bool.true_eq, ite_true, mul_one] |
| rw [abs_div, abs_of_pos hb] |
| apply (div_le_iff₀ hb).mpr |
| linarith |
| · ext j |
| fin_cases j <;> simp [mapCenteredToCanonical, upperPoint, point, height, |
| effectiveTilt] <;> field_simp |
| · rintro ⟨r, hr, rho, hrho, x, hx, hz⟩ |
| refine ⟨upperPoint b eta rho (b*r) (b*x), ?_, ?_⟩ |
| · refine ⟨b*r, ?_, rho, hrho, (b*x), ?_, rfl⟩ |
| · apply (RadialRiemann.div_mem_symmetric_cell_iff hb N i (b*r)).1 |
| simpa [hb0] using hr |
| · norm_num only [axialBound, Bool.true_eq, ite_true, mul_one] at hx |
| change |(b*x)| ≤ 2*b |
| rw [abs_mul, abs_of_pos hb] |
| nlinarith |
| · rw [hz] |
| ext j |
| fin_cases j <;> simp [mapCenteredToCanonical, upperPoint, point, height, |
| effectiveTilt] <;> field_simp |
|
|
| lemma centered_lowerLocal_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (i : Fin N) : |
| mapCenteredToCanonical b '' lowerSweepOn true b eta T (RadialRiemann.cell (-b) b N i) = |
| lowerSweepOn true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| (RadialRiemann.cell (-1) 1 N i) := by |
| have hb0 := ne_of_gt hb |
| ext z |
| constructor |
| · rintro ⟨v, ⟨s, hs, rho, hrho, x, hx, rfl⟩, rfl⟩ |
| have hbs : b*(s/b)=s := by field_simp |
| refine ⟨s/b, (RadialRiemann.div_mem_symmetric_cell_iff hb N i s).2 hs, |
| rho, ⟨hrho.1, ?_⟩, x/b, ?_, ?_⟩ |
| · change rho ≤ T (b*(s/b)) |
| rw [hbs] |
| linarith [hrho.2] |
| · change |x| ≤ 2*b at hx |
| norm_num only [axialBound, Bool.true_eq, ite_true, mul_one] |
| rw [abs_div, abs_of_pos hb] |
| apply (div_le_iff₀ hb).mpr |
| linarith |
| · ext j |
| fin_cases j <;> simp [mapCenteredToCanonical, lowerPoint, point, height, |
| effectiveTilt] <;> field_simp |
| · rintro ⟨r, hr, rho, hrho, x, hx, hz⟩ |
| refine ⟨lowerPoint true b eta rho (b*r) (b*x), ?_, ?_⟩ |
| · refine ⟨b*r, ?_, rho, hrho, (b*x), ?_, rfl⟩ |
| · apply (RadialRiemann.div_mem_symmetric_cell_iff hb N i (b*r)).1 |
| simpa [hb0] using hr |
| · norm_num only [axialBound, Bool.true_eq, ite_true, mul_one] at hx |
| change |(b*x)| ≤ 2*b |
| rw [abs_mul, abs_of_pos hb] |
| nlinarith |
| · rw [hz] |
| ext j |
| fin_cases j <;> simp [mapCenteredToCanonical, lowerPoint, point, height, |
| effectiveTilt] <;> field_simp |
|
|
| lemma upperPhysicalAxis_normalized {b : ℝ} (hb : 0 < b) (eta s : ℝ) : |
| upperPhysicalAxis b eta s = |
| upperPhysicalAxis 1 (effectiveTilt b eta) (s/b) := by |
| rw [upperPhysicalAxis_canonical (ne_of_gt hb), |
| upperPhysicalAxis_canonical (show (1 : ℝ) ≠ 0 by norm_num)] |
| simp [effectiveTilt] |
|
|
| lemma lowerPhysicalAxis_normalized {b : ℝ} (hb : 0 < b) (eta s : ℝ) : |
| lowerPhysicalAxis b eta s = |
| lowerPhysicalAxis 1 (effectiveTilt b eta) (s/b) := by |
| rw [lowerPhysicalAxis_canonical (ne_of_gt hb), |
| lowerPhysicalAxis_canonical (show (1 : ℝ) ≠ 0 by norm_num)] |
| simp [effectiveTilt] |
|
|
| lemma halfheight_upperBase_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| (fun z => (2/b) • z - |
| TriangularCovering.projection (upperPhysicalAxis b eta (tag i)) |
| (!₂[0,0,Real.sqrt 2] : Space)) '' |
| upperPhysicalBase false b eta T N tag i = |
| upperPhysicalBase true 1 (effectiveTilt b eta) (fun r => 2*T (b*r)) |
| N (fun j => tag j/b) i := by |
| unfold upperPhysicalBase |
| rw [← projected_image_mapToCanonical, halfheight_upperLocal_image hb, |
| upperPhysicalAxis_normalized hb] |
|
|
| lemma halfheight_upperBase_area {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| TriangularCovering.area |
| (upperPhysicalBase true 1 (effectiveTilt b eta) (fun r => 2*T (b*r)) |
| N (fun j => tag j/b) i) = |
| (2/b)^2 * TriangularCovering.area (upperPhysicalBase false b eta T N tag i) := by |
| unfold upperPhysicalBase |
| rw [← upperPhysicalAxis_normalized hb eta (tag i), |
| ← halfheight_upperLocal_image hb eta T N i] |
| exact projected_area_mapToCanonical (ne_of_gt hb) _ _ |
|
|
| lemma halfheight_lowerBase_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| (fun z => (2/b) • z - |
| TriangularCovering.projection (lowerPhysicalAxis b eta (tag i)) |
| (!₂[0,0,Real.sqrt 2] : Space)) '' |
| lowerPhysicalBase false b eta T N tag i = |
| lowerPhysicalBase true 1 (effectiveTilt b eta) (fun r => 2*T (b*r)) |
| N (fun j => tag j/b) i := by |
| unfold lowerPhysicalBase |
| rw [← projected_image_mapToCanonical, halfheight_lowerLocal_image hb, |
| lowerPhysicalAxis_normalized hb] |
|
|
| lemma halfheight_lowerBase_area {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| TriangularCovering.area |
| (lowerPhysicalBase true 1 (effectiveTilt b eta) (fun r => 2*T (b*r)) |
| N (fun j => tag j/b) i) = |
| (2/b)^2 * TriangularCovering.area (lowerPhysicalBase false b eta T N tag i) := by |
| unfold lowerPhysicalBase |
| rw [← lowerPhysicalAxis_normalized hb eta (tag i), |
| ← halfheight_lowerLocal_image hb eta T N i] |
| exact projected_area_mapToCanonical (ne_of_gt hb) _ _ |
|
|
| lemma centered_upperBase_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| (fun z => (1/b) • z) '' |
| upperPhysicalBase true b eta T N tag i = |
| upperPhysicalBase true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| N (fun j => tag j/b) i := by |
| unfold upperPhysicalBase |
| rw [← projected_image_mapCenteredToCanonical, centered_upperLocal_image hb, |
| upperPhysicalAxis_normalized hb] |
|
|
| lemma centered_upperBase_area {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| TriangularCovering.area |
| (upperPhysicalBase true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| N (fun j => tag j/b) i) = |
| (1/b)^2 * TriangularCovering.area (upperPhysicalBase true b eta T N tag i) := by |
| unfold upperPhysicalBase |
| rw [← upperPhysicalAxis_normalized hb eta (tag i), |
| ← centered_upperLocal_image hb eta T N i] |
| exact projected_area_mapCenteredToCanonical (ne_of_gt hb) _ _ |
|
|
| lemma centered_lowerBase_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| (fun z => (1/b) • z) '' |
| lowerPhysicalBase true b eta T N tag i = |
| lowerPhysicalBase true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| N (fun j => tag j/b) i := by |
| unfold lowerPhysicalBase |
| rw [← projected_image_mapCenteredToCanonical, centered_lowerLocal_image hb, |
| lowerPhysicalAxis_normalized hb] |
|
|
| lemma centered_lowerBase_area {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| TriangularCovering.area |
| (lowerPhysicalBase true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| N (fun j => tag j/b) i) = |
| (1/b)^2 * TriangularCovering.area (lowerPhysicalBase true b eta T N tag i) := by |
| unfold lowerPhysicalBase |
| rw [← lowerPhysicalAxis_normalized hb eta (tag i), |
| ← centered_lowerLocal_image hb eta T N i] |
| exact projected_area_mapCenteredToCanonical (ne_of_gt hb) _ _ |
|
|
| lemma halfheight_actualCost_scale {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tagU tagL : Fin N → ℝ) : |
| physicalActualCost true 1 (effectiveTilt b eta) (fun r => 2*T (b*r)) |
| N (fun j => tagU j/b) (fun j => tagL j/b) = |
| (2/b)^2 * physicalActualCost false b eta T N tagU tagL := by |
| unfold physicalActualCost |
| simp only [halfheight_upperBase_area hb, halfheight_lowerBase_area hb, |
| ← Finset.mul_sum, mul_add] |
|
|
| lemma centered_actualCost_scale {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tagU tagL : Fin N → ℝ) : |
| physicalActualCost true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| N (fun j => tagU j/b) (fun j => tagL j/b) = |
| (1/b)^2 * physicalActualCost true b eta T N tagU tagL := by |
| unfold physicalActualCost |
| simp only [centered_upperBase_area hb, centered_lowerBase_area hb, |
| ← Finset.mul_sum, mul_add] |
|
|
| lemma halfheight_upperCylinder_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| mapToCanonical b '' |
| RadialSweep.cylinder (upperPhysicalAxis b eta (tag i)) |
| (upperPhysicalBase false b eta T N tag i) = |
| RadialSweep.cylinder (upperPhysicalAxis 1 (effectiveTilt b eta) (tag i/b)) |
| (upperPhysicalBase true 1 (effectiveTilt b eta) (fun r => 2*T (b*r)) |
| N (fun j => tag j/b) i) := by |
| rw [cylinder_mapToCanonical (ne_of_gt hb), halfheight_upperBase_image hb, |
| upperPhysicalAxis_normalized hb] |
|
|
| lemma halfheight_lowerCylinder_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| mapToCanonical b '' |
| RadialSweep.cylinder (lowerPhysicalAxis b eta (tag i)) |
| (lowerPhysicalBase false b eta T N tag i) = |
| RadialSweep.cylinder (lowerPhysicalAxis 1 (effectiveTilt b eta) (tag i/b)) |
| (lowerPhysicalBase true 1 (effectiveTilt b eta) (fun r => 2*T (b*r)) |
| N (fun j => tag j/b) i) := by |
| rw [cylinder_mapToCanonical (ne_of_gt hb), halfheight_lowerBase_image hb, |
| lowerPhysicalAxis_normalized hb] |
|
|
| lemma centered_upperCylinder_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| mapCenteredToCanonical b '' |
| RadialSweep.cylinder (upperPhysicalAxis b eta (tag i)) |
| (upperPhysicalBase true b eta T N tag i) = |
| RadialSweep.cylinder (upperPhysicalAxis 1 (effectiveTilt b eta) (tag i/b)) |
| (upperPhysicalBase true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| N (fun j => tag j/b) i) := by |
| rw [cylinder_mapCenteredToCanonical (ne_of_gt hb), centered_upperBase_image hb, |
| upperPhysicalAxis_normalized hb] |
|
|
| lemma centered_lowerCylinder_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ) |
| (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) : |
| mapCenteredToCanonical b '' |
| RadialSweep.cylinder (lowerPhysicalAxis b eta (tag i)) |
| (lowerPhysicalBase true b eta T N tag i) = |
| RadialSweep.cylinder (lowerPhysicalAxis 1 (effectiveTilt b eta) (tag i/b)) |
| (lowerPhysicalBase true 1 (effectiveTilt b eta) (fun r => T (b*r)) |
| N (fun j => tag j/b) i) := by |
| rw [cylinder_mapCenteredToCanonical (ne_of_gt hb), centered_lowerBase_image hb, |
| lowerPhysicalAxis_normalized hb] |
|
|
| end RadialPhysical |
|
|
| end |
|
|
| end OAI |
|
|
