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openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Physical /Triangles.lean
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import OAI.Geometry.CylinderCovering.Physical.TrianglesDefinitions
import OAI.Geometry.CylinderCovering.Physical.Finite
namespace OAI
open Set MeasureTheory Filter
open scoped Topology InnerProductSpace
noncomputable section
namespace RadialPhysical.ExplicitTriangles
open CylinderCovering PhysicalFiniteProof
variable (centered : Bool) (b eta : ℝ) (hb : 0 < b) (T : ℝ → ℝ)
(hT : ContinuousOn T (Icc (-b) b))
(hT0 : ∀ s ∈ Icc (-b) b, 0 ≤ T s)
lemma upper_scalarBound :
RadialSweep.scalarBound (upperData centered b eta hb T hT hT0) = |eta| := by
unfold RadialSweep.scalarBound
change sSup ((fun _ : ℝ => |-eta|) '' Icc (-b) b) = |eta|
have him : ((fun _ : ℝ => |-eta|) '' Icc (-b) b) = {|eta|} := by
ext z
constructor
· rintro ⟨s, hs, rfl⟩
simp
· rintro rfl
exact ⟨0, ⟨by linarith, by linarith⟩, abs_neg eta⟩
rw [him, csSup_singleton]
lemma lower_scalarBound :
RadialSweep.scalarBound (lowerData centered b eta hb T hT hT0) = |eta| := by
unfold RadialSweep.scalarBound
change sSup ((fun _ : ℝ => |eta|) '' Icc (-b) b) = |eta|
have him : ((fun _ : ℝ => |eta|) '' Icc (-b) b) = {|eta|} := by
ext z
constructor
· rintro ⟨s, hs, rfl⟩
rfl
· rintro rfl
exact ⟨0, ⟨by linarith, by linarith⟩, rfl⟩
rw [him, csSup_singleton]
lemma upper_error :
RadialSweep.radialError (upperData centered b eta hb T hT hT0) =
error centered b eta := by
unfold RadialSweep.radialError
rw [upper_scalarBound]
rfl
lemma lower_error :
RadialSweep.radialError (lowerData centered b eta hb T hT hT0) =
error centered b eta := by
unfold RadialSweep.radialError
rw [lower_scalarBound]
rfl
lemma upper_coordinateMap (p : ℝ × ℝ) :
RadialSweep.coordinateMap (upperData centered b eta hb T hT hT0) p =
upperIntercept b p := by
ext i
fin_cases i <;> simp [RadialSweep.coordinateMap, upperData, upperIntercept,
point, RadialPairedSweep.point]
ring
lemma lower_coordinateMap (p : ℝ × ℝ) :
RadialSweep.coordinateMap (lowerData centered b eta hb T hT hT0) p =
lowerIntercept centered b p := by
cases centered <;> ext i <;>
fin_cases i <;> simp [RadialSweep.coordinateMap, lowerData, lowerIntercept,
point, RadialPairedSweep.point] <;> ring
lemma upper_vertices (N : ℕ) (i : Fin N) :
RadialSweep.frozenVertices (upperData centered b eta hb T hT hT0) N i =
upperVertices centered b eta T N i := by
unfold RadialSweep.frozenVertices
simp only [RadialSweep.leftTip, RadialSweep.rightEdge, RadialSweep.collar,
upper_error, RadialSweep.midpoint]
change ![_, _, _] = _
funext j
fin_cases j <;>
change RadialSweep.coordinateMap (upperData centered b eta hb T hT hT0) _ =
upperIntercept b _
all_goals
rw [upper_coordinateMap]
rfl
lemma lower_vertices (N : ℕ) (i : Fin N) :
RadialSweep.frozenVertices (lowerData centered b eta hb T hT hT0) N i =
lowerVertices centered b eta T N i := by
unfold RadialSweep.frozenVertices
simp only [RadialSweep.leftTip, RadialSweep.rightEdge, RadialSweep.collar,
lower_error, RadialSweep.midpoint]
change ![_, _, _] = _
funext j
fin_cases j <;>
change RadialSweep.coordinateMap (lowerData centered b eta hb T hT hT0) _ =
lowerIntercept centered b _
all_goals
rw [lower_coordinateMap]
rfl
lemma upper_base (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) :
RadialSweep.frozenBase (upperData centered b eta hb T hT hT0) N tag i =
upperBase centered b eta T N tag i := by
unfold RadialSweep.frozenBase RadialSweep.frozenTriangle upperBase
rw [upperData_direction, upper_vertices]
lemma lower_base (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) :
RadialSweep.frozenBase (lowerData centered b eta hb T hT hT0) N tag i =
lowerBase centered b eta T N tag i := by
unfold RadialSweep.frozenBase RadialSweep.frozenTriangle lowerBase
rw [lowerData_direction, lower_vertices]
lemma upper_speed (s : ℝ) :
‖(upperData centered b eta hb T hT hT0).w s‖^2 =
eta^2 * ((s^2+b^2)^2/4 + height b^2*s^2) := by
have hv : (upperData centered b eta hb T hT hT0).w s =
RadialPairedSweep.point 0 (-eta/2*(s^2+b^2)) (height b*eta*s) := by
change (-eta/2*(s^2+b^2)) • RadialPairedSweep.point 0 1 0 +
(height b*eta*s) • RadialPairedSweep.point 0 0 1 = _
ext i
fin_cases i <;> simp [RadialPairedSweep.point]
rw [hv, RadialPairedSweep.point_norm_sq]
ring
lemma lower_speed (s : ℝ) :
‖(lowerData centered b eta hb T hT hT0).w s‖^2 =
eta^2 * ((s^2+b^2)^2/4 + height b^2*s^2) := by
have hv : (lowerData centered b eta hb T hT hT0).w s =
RadialPairedSweep.point (eta/2*(s^2+b^2)) 0 (height b*eta*s) := by
change (eta/2*(s^2+b^2)) • RadialPairedSweep.point 1 0 0 +
(height b*eta*s) • RadialPairedSweep.point 0 0 1 = _
ext i
fin_cases i <;> simp [RadialPairedSweep.point]
rw [hv, RadialPairedSweep.point_norm_sq]
ring
lemma upper_area (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) :
RadialSweep.frozenArea (upperData centered b eta hb T hT hT0) N tag i =
localArea centered b eta T N tag i := by
unfold RadialSweep.frozenArea RadialSweep.collar
rw [upperData_kappa, upper_error, upper_speed]
change height b * RadialRiemann.step (-b) b N *
(RadialRiemann.capMax T (-b) b N i +
(error centered b eta + (3*error centered b eta+1))*RadialRiemann.step (-b) b N)^2 /
(2*Real.sqrt (1+eta^2*((tag i^2+b^2)^2/4+height b^2*tag i^2))) = _
unfold localArea
ring
lemma lower_area (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) :
RadialSweep.frozenArea (lowerData centered b eta hb T hT hT0) N tag i =
localArea centered b eta T N tag i := by
unfold RadialSweep.frozenArea RadialSweep.collar
rw [lowerData_kappa, lower_error, lower_speed]
change height b * RadialRiemann.step (-b) b N *
(RadialRiemann.capMax T (-b) b N i +
(error centered b eta + (3*error centered b eta+1))*RadialRiemann.step (-b) b N)^2 /
(2*Real.sqrt (1+eta^2*((tag i^2+b^2)^2/4+height b^2*tag i^2))) = _
unfold localArea
ring
def upperCylinder (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
TriangularCovering.TriCylinder :=
RadialSweep.frozenCylinder (upperData centered b eta hb T hT hT0) N tag htag i
def lowerCylinder (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
TriangularCovering.TriCylinder :=
RadialSweep.frozenCylinder (lowerData centered b eta hb T hT hT0) N tag htag i
theorem upperCylinder_axis (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
(upperCylinder centered b eta hb T hT hT0 N tag htag i).axis =
upperPhysicalAxis b eta (tag i) :=
upperData_direction centered b eta hb T hT hT0 (tag i)
theorem lowerCylinder_axis (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
(lowerCylinder centered b eta hb T hT hT0 N tag htag i).axis =
lowerPhysicalAxis b eta (tag i) :=
lowerData_direction centered b eta hb T hT hT0 (tag i)
theorem upperCylinder_base (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
(upperCylinder centered b eta hb T hT hT0 N tag htag i).base =
upperBase centered b eta T N tag i :=
upper_base centered b eta hb T hT hT0 N tag i
theorem lowerCylinder_base (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
(lowerCylinder centered b eta hb T hT hT0 N tag htag i).base =
lowerBase centered b eta T N tag i :=
lower_base centered b eta hb T hT hT0 N tag i
theorem upperCylinder_area (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
TriangularCovering.area
(upperCylinder centered b eta hb T hT hT0 N tag htag i).base =
localArea centered b eta T N tag i := by
exact (RadialSweep.frozenCylinder_area
(upperData centered b eta hb T hT hT0) N tag htag i).trans
(upper_area centered b eta hb T hT hT0 N tag i)
theorem lowerCylinder_area (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
TriangularCovering.area
(lowerCylinder centered b eta hb T hT hT0 N tag htag i).base =
localArea centered b eta T N tag i := by
exact (RadialSweep.frozenCylinder_area
(lowerData centered b eta hb T hT hT0) N tag htag i).trans
(lower_area centered b eta hb T hT hT0 N tag i)
theorem upper_actual_enclosure (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
upperPhysicalBase centered b eta T N tag i ⊆
(upperCylinder centered b eta hb T hT hT0 N tag htag i).base := by
rw [upperCylinder_base, ← upper_base centered b eta hb T hT hT0,
← upperData_actualBase centered b eta hb T hT hT0]
exact RadialSweep.actualBase_subset_frozenBase _ tag htag i
theorem lower_actual_enclosure (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
lowerPhysicalBase centered b eta T N tag i ⊆
(lowerCylinder centered b eta hb T hT hT0 N tag htag i).base := by
rw [lowerCylinder_base, ← lower_base centered b eta hb T hT hT0,
← lowerData_actualBase centered b eta hb T hT hT0]
exact RadialSweep.actualBase_subset_frozenBase _ tag htag i
theorem upper_cover {N : ℕ} (hN : 0 < N) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) :
∀ z ∈ upperSweepOn centered b eta T (Icc (-b) b),
∃ i : Fin N, z ∈ (upperCylinder centered b eta hb T hT hT0 N tag htag i).carrier := by
intro z hz
rw [← upperData_sweep centered b eta hb T hT hT0,
RadialSweep.sweep_eq_iUnion_localSweep _ hN] at hz
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hz
exact ⟨i, RadialSweep.localSweep_mem_frozenCylinder _ tag htag i hi⟩
theorem lower_cover {N : ℕ} (hN : 0 < N) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) :
∀ z ∈ lowerSweepOn centered b eta T (Icc (-b) b),
∃ i : Fin N, z ∈ (lowerCylinder centered b eta hb T hT hT0 N tag htag i).carrier := by
intro z hz
rw [← lowerData_sweep centered b eta hb T hT hT0,
RadialSweep.sweep_eq_iUnion_localSweep _ hN] at hz
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hz
exact ⟨i, RadialSweep.localSweep_mem_frozenCylinder _ tag htag i hi⟩
theorem upper_total_tendsto (tag : (N : ℕ) → Fin N → ℝ)
(htag : ∀ N, RadialRiemann.IsTagged (-b) b N (tag N)) :
Tendsto (fun N => TriangularCovering.totalArea
(upperCylinder centered b eta hb T hT hT0 N (tag N) (htag N)))
atTop (𝓝 (scaledCost b eta T / 2)) := by
have h := RadialSweep.frozenArea_tendsto
(upperData centered b eta hb T hT hT0) tag htag
rw [upperData_weightedArea] at h
convert h using 1
funext N
apply Finset.sum_congr rfl
intro i _
exact RadialSweep.frozenCylinder_area _ N (tag N) (htag N) i
theorem lower_total_tendsto (tag : (N : ℕ) → Fin N → ℝ)
(htag : ∀ N, RadialRiemann.IsTagged (-b) b N (tag N)) :
Tendsto (fun N => TriangularCovering.totalArea
(lowerCylinder centered b eta hb T hT hT0 N (tag N) (htag N)))
atTop (𝓝 (scaledCost b eta T / 2)) := by
have h := RadialSweep.frozenArea_tendsto
(lowerData centered b eta hb T hT hT0) tag htag
rw [lowerData_weightedArea] at h
convert h using 1
funext N
apply Finset.sum_congr rfl
intro i _
exact RadialSweep.frozenCylinder_area _ N (tag N) (htag N) i
theorem total_tendsto (tagU tagL : (N : ℕ) → Fin N → ℝ)
(htagU : ∀ N, RadialRiemann.IsTagged (-b) b N (tagU N))
(htagL : ∀ N, RadialRiemann.IsTagged (-b) b N (tagL N)) :
Tendsto (fun N => TriangularCovering.totalArea
(upperCylinder centered b eta hb T hT hT0 N (tagU N) (htagU N)) +
TriangularCovering.totalArea
(lowerCylinder centered b eta hb T hT hT0 N (tagL N) (htagL N)))
atTop (𝓝 (scaledCost b eta T)) := by
simpa only [add_halves] using
(upper_total_tendsto centered b eta hb T hT hT0 tagU htagU).add
(lower_total_tendsto centered b eta hb T hT hT0 tagL htagL)
include hb hT hT0 in
theorem localArea_pos (N : ℕ) (tag : Fin N → ℝ) (i : Fin N) :
0 < localArea centered b eta T N tag i := by
have hinterval : -b < b := by linarith
have hd := RadialRiemann.step_pos hinterval (Nat.zero_lt_of_lt i.isLt)
have hcap := RadialRiemann.capMax_nonneg hinterval hT hT0 i
have hh : 0 < height b := mul_pos (Real.sqrt_pos.mpr (by norm_num)) hb
have hA : 0 ≤ error centered b eta := by
unfold error axialBound
cases centered <;> simp <;> positivity
have hwidth : 0 < RadialRiemann.capMax T (-b) b N i +
(4 * error centered b eta + 1) * RadialRiemann.step (-b) b N := by positivity
unfold localArea
exact div_pos (mul_pos (mul_pos hh hd) (sq_pos_of_pos hwidth))
(mul_pos (by norm_num) (Real.sqrt_pos.mpr (by positivity)))
theorem physical_triangle_construction : ConstructionStatement := by
intro centered b eta hb T hT hT0 tagU tagL htagU htagL
refine ⟨(fun N => upperCylinder centered b eta hb T hT hT0 N (tagU N) (htagU N)),
(fun N => lowerCylinder centered b eta hb T hT hT0 N (tagL N) (htagL N)), ?_, ?_, ?_⟩
· intro N i
exact ⟨upperCylinder_axis centered b eta hb T hT hT0 N (tagU N) (htagU N) i,
lowerCylinder_axis centered b eta hb T hT hT0 N (tagL N) (htagL N) i,
upperCylinder_base centered b eta hb T hT hT0 N (tagU N) (htagU N) i,
lowerCylinder_base centered b eta hb T hT hT0 N (tagL N) (htagL N) i,
upperCylinder_area centered b eta hb T hT hT0 N (tagU N) (htagU N) i,
lowerCylinder_area centered b eta hb T hT hT0 N (tagL N) (htagL N) i,
upper_actual_enclosure centered b eta hb T hT hT0 N (tagU N) (htagU N) i,
lower_actual_enclosure centered b eta hb T hT hT0 N (tagL N) (htagL N) i⟩
· intro N hN
exact ⟨upper_cover centered b eta hb T hT hT0 hN (tagU N) (htagU N),
lower_cover centered b eta hb T hT hT0 hN (tagL N) (htagL N)⟩
· exact total_tendsto centered b eta hb T hT hT0 tagU tagL htagU htagL
include hb hT hT0 in
theorem upperActual_area_le (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
TriangularCovering.area (upperPhysicalBase centered b eta T N tag i) ≤
localArea centered b eta T N tag i := by
have h := measureReal_mono
(upper_actual_enclosure centered b eta hb T hT hT0 N tag htag i)
(upperCylinder centered b eta hb T hT hT0 N tag htag i).finite_area
change TriangularCovering.area _ ≤ TriangularCovering.area _ at h
rwa [upperCylinder_area] at h
include hb hT hT0 in
theorem lowerActual_area_le (N : ℕ) (tag : Fin N → ℝ)
(htag : RadialRiemann.IsTagged (-b) b N tag) (i : Fin N) :
TriangularCovering.area (lowerPhysicalBase centered b eta T N tag i) ≤
localArea centered b eta T N tag i := by
have h := measureReal_mono
(lower_actual_enclosure centered b eta hb T hT hT0 N tag htag i)
(lowerCylinder centered b eta hb T hT hT0 N tag htag i).finite_area
change TriangularCovering.area _ ≤ TriangularCovering.area _ at h
rwa [lowerCylinder_area] at h
include hb hT hT0 in
theorem actualCost_le_formula (N : ℕ) (tagU tagL : Fin N → ℝ)
(htagU : RadialRiemann.IsTagged (-b) b N tagU)
(htagL : RadialRiemann.IsTagged (-b) b N tagL) :
physicalActualCost centered b eta T N tagU tagL ≤
(∑ i : Fin N, localArea centered b eta T N tagU i) +
(∑ i : Fin N, localArea centered b eta T N tagL i) := by
apply add_le_add
· exact Finset.sum_le_sum fun i _ =>
upperActual_area_le centered b eta hb T hT hT0 N tagU htagU i
· exact Finset.sum_le_sum fun i _ =>
lowerActual_area_le centered b eta hb T hT hT0 N tagL htagL i
end RadialPhysical.ExplicitTriangles
end
end OAI