| import Mathlib |
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| namespace OAI |
|
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| open scoped BigOperators ContDiff |
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| namespace BoxTransport.Routing |
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|
| abbrev Space := Fin 3 → ℝ |
| abbrev RationalSpace := Fin 3 → ℚ |
| abbrev SpaceTime := ℝ × Space |
| abbrev RationalSpaceTime := ℚ × RationalSpace |
| abbrev Field := SpaceTime → Space |
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|
| abbrev Axis := Option (Fin 3) |
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| def coordinateVector (j : Fin 3) : Space := Pi.single j 1 |
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|
| def coordinateDirection : Axis → SpaceTime |
| | none => (1, 0) |
| | some j => (0, coordinateVector j) |
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| def realPoint (q : RationalSpace) : Space := fun j => (q j : ℝ) |
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| def realSpaceTime (q : RationalSpaceTime) : SpaceTime := (q.1, realPoint q.2) |
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| def realBox (c r : Space) : Set Space := |
| {x | ∀ j, |x j - c j| ≤ r j} |
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| def solidBox (a h : RationalSpace) : Set Space := |
| realBox (realPoint a) (realPoint h) |
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| def prescribedAffine (a b h k : RationalSpace) (x : Space) : Space := |
| fun j => (b j : ℝ) + ((k j / h j : ℚ) : ℝ) * (x j - (a j : ℝ)) |
|
|
| noncomputable def mixedDerivative : List Axis → Field → Field |
| | [], U => U |
| | j :: js, U => fun p => fderiv ℝ (mixedDerivative js U) p (coordinateDirection j) |
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| abbrev RationalCode := ℕ × ℕ × ℕ |
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| def rationalValue (q : RationalCode) : ℚ := ((q.1 : ℚ) - q.2.1) / (q.2.2 + 1) |
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| abbrev RationalPointCode := RationalCode × RationalCode × RationalCode × RationalCode |
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| def codeSpaceTime (q : RationalPointCode) : SpaceTime := |
| ((rationalValue q.1 : ℝ), ![(rationalValue q.2.1 : ℝ), |
| (rationalValue q.2.2.1 : ℝ), (rationalValue q.2.2.2 : ℝ)]) |
|
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| abbrev EvaluationRequest := List Axis × RationalPointCode × ℕ × Fin 3 |
|
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| def Effective (U : Field) : Prop := |
| ∃ eval : EvaluationRequest → RationalCode, |
| ∃ modulus : List Axis × ℕ → ℕ, |
| ∃ bound : List Axis → ℕ, |
| Computable eval ∧ Computable modulus ∧ Computable bound ∧ |
| (∀ (ds : List Axis) (q : RationalPointCode) (n : ℕ) (p : SpaceTime), |
| ‖p - codeSpaceTime q‖ ≤ ((modulus (ds, n) : ℝ) + 1)⁻¹ → |
| ∀ j : Fin 3, |
| |mixedDerivative ds U p j - (rationalValue (eval (ds, q, n, j)) : ℝ)| ≤ |
| ((n : ℝ) + 1)⁻¹) ∧ |
| (∀ (ds : List Axis) (p : SpaceTime) (j : Fin 3), |
| |mixedDerivative ds U p j| ≤ (bound ds : ℝ)) |
|
|
| noncomputable def spatialDivergence (U : Field) (t : ℝ) (x : Space) : ℝ := |
| ∑ j : Fin 3, (fderiv ℝ (fun y => U (t, y)) x (coordinateVector j)) j |
|
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| def AdmissibleField (U : Field) : Prop := |
| ContDiff ℝ ∞ U ∧ HasCompactSupport U ∧ Effective U ∧ |
| (∀ t x, spatialDivergence U t x = 0) ∧ |
| (∀ (t : ℝ) (x : Space), t ∉ Set.Icc (1 / 4 : ℝ) (3 / 4 : ℝ) → U (t, x) = 0) |
|
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| def IsGlobalFlow (U : Field) (Φ : ℝ → Space → Space) : Prop := |
| (∀ x, Φ 0 x = x) ∧ |
| (∀ t x, HasDerivAt (fun s => Φ s x) (U (t, Φ t x)) t) ∧ |
| (∀ (x : Space) (γ : ℝ → Space), γ 0 = x → |
| (∀ t, HasDerivAt γ (U (t, γ t)) t) → ∀ t, γ t = Φ t x) |
|
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| def Delivers {n : ℕ} (a b h k : Fin n → RationalSpace) |
| (Φ : ℝ → Space → Space) : Prop := |
| ∀ i, ∃ O : Set Space, IsOpen O ∧ solidBox (a i) (h i) ⊆ O ∧ |
| ∀ x ∈ O, Φ 1 x = prescribedAffine (a i) (b i) (h i) (k i) x |
|
|
| theorem balanced_box_routing |
| (n : ℕ) (a b h k : Fin n → RationalSpace) |
| (hh : ∀ i j, 0 < h i j) (hk : ∀ i j, 0 < k i j) |
| (ha : Pairwise fun i j => Disjoint (solidBox (a i) (h i)) (solidBox (a j) (h j))) |
| (hb : Pairwise fun i j => Disjoint (solidBox (b i) (k i)) (solidBox (b j) (k j))) |
| (hvol : ∀ i, ∏ j : Fin 3, k i j / h i j = 1) : |
| ∃ U : Field, ∃ Φ : ℝ → Space → Space, |
| AdmissibleField U ∧ IsGlobalFlow U Φ ∧ Delivers a b h k Φ := by |
| sorry |
|
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| end BoxTransport.Routing |
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| end OAI |
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|
