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import OAI.Geometry.CylinderCovering.Radial.PartitionAffine
namespace OAI
open Set
noncomputable section
namespace RadialPhysical
open CylinderCovering
lemma halfheight_upperLocal_image {b : ℝ} (hb : 0 < b) (eta : ℝ) (T : ℝ → ℝ)
(N : ℕ) (i : Fin N) :
mapToCanonical b '' upperSweepOn false b eta T (RadialRiemann.cell (-b) b N i) =
upperSweepOn 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, 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
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