openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /FiniteParameterChain.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.FiveGainPullback | |
| namespace OAI | |
| /-! # A finite parameterized gain chain as one product integral | |
| This is the Fubini assembly used after replacing each physical collision by its | |
| thermal and redundant-plane parameters. The transition maps and densities are | |
| explicit functions, and the conclusion is an exact integral identity. | |
| -/ | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace BoltzmannNonuniqueness.FiveGainMap | |
| def parameterChain (next : ℕ → State → StageControls → State) | |
| (density : ℕ → State → StageControls → ℝ≥0∞) (g : State → ℝ≥0∞) | |
| (i : ℕ) : ℕ → State → ℝ≥0∞ | |
| | 0, z => g z | |
| | n+1, z => ∫⁻ p : StageControls, density i z p* | |
| parameterChain next density g (i+1) n (next i z p) | |
| def chainIntegrand (next : ℕ → State → StageControls → State) | |
| (density : ℕ → State → StageControls → ℝ≥0∞) (g : State → ℝ≥0∞) | |
| (i : ℕ) : (n : ℕ) → State → (Fin n → StageControls) → ℝ≥0∞ | |
| | 0, z, _ => g z | |
| | n+1, z, q => density i z (q 0)* | |
| chainIntegrand next density g (i+1) n (next i z (q 0)) (Fin.tail q) | |
| theorem integrateControls_const_mul {c : ℝ≥0∞} (hc : c ≠ ∞) (n : ℕ) | |
| (F : (Fin n → StageControls) → ℝ≥0∞) : | |
| integrateControls n (fun q => c*F q) = c*integrateControls n F := by | |
| induction n with | |
| | zero => rfl | |
| | succ n ih => | |
| simp only [integrateControls,ih] | |
| exact lintegral_const_mul' _ _ hc | |
| theorem parameterChain_eq_nested (next : ℕ → State → StageControls → State) | |
| (density : ℕ → State → StageControls → ℝ≥0∞) | |
| (hfinite : ∀ i z p, density i z p ≠ ∞) (g : State → ℝ≥0∞) (i n : ℕ) (z : State) : | |
| parameterChain next density g i n z = | |
| integrateControls n (chainIntegrand next density g i n z) := by | |
| induction n generalizing i z with | |
| | zero => rfl | |
| | succ n ih => | |
| simp only [parameterChain,integrateControls] | |
| apply lintegral_congr | |
| intro p | |
| rw [ih] | |
| have he := integrateControls_const_mul (hfinite i z p) n | |
| (chainIntegrand next density g (i+1) n (next i z p)) | |
| simpa only [chainIntegrand,Fin.cons_zero,Fin.tail_cons] using he.symm | |
| theorem chainIntegrand_measurable (next : ℕ → State → StageControls → State) | |
| (density : ℕ → State → StageControls → ℝ≥0∞) | |
| (hnext : ∀ i, Measurable (fun p : State × StageControls => next i p.1 p.2)) | |
| (hdensity : ∀ i, Measurable (fun p : State × StageControls => density i p.1 p.2)) | |
| {g : State → ℝ≥0∞} (hg : Measurable g) (i n : ℕ) : | |
| Measurable (fun p : State × (Fin n → StageControls) => | |
| chainIntegrand next density g i n p.1 p.2) := by | |
| induction n generalizing i with | |
| | zero => exact hg.comp measurable_fst | |
| | succ n ih => | |
| have hfirst : Measurable (fun p : State × (Fin (n+1) → StageControls) => (p.1,p.2 0)) := | |
| measurable_fst.prodMk ((measurable_pi_apply 0).comp measurable_snd) | |
| have htail : Measurable (fun p : State × (Fin (n+1) → StageControls) => Fin.tail p.2) := by | |
| exact Measurable.of_eval (fun j => (measurable_pi_apply j.succ).comp measurable_snd) | |
| exact ((hdensity i).comp hfirst).mul | |
| ((ih (i+1)).comp (((hnext i).comp hfirst).prodMk htail)) | |
| theorem parameterChain_eq_product (next : ℕ → State → StageControls → State) | |
| (density : ℕ → State → StageControls → ℝ≥0∞) | |
| (hnext : ∀ i, Measurable (fun p : State × StageControls => next i p.1 p.2)) | |
| (hdensity : ∀ i, Measurable (fun p : State × StageControls => density i p.1 p.2)) | |
| (hfinite : ∀ i z p, density i z p ≠ ∞) {g : State → ℝ≥0∞} (hg : Measurable g) | |
| (i n : ℕ) (z : State) : | |
| parameterChain next density g i n z = | |
| ∫⁻ q : Fin n → StageControls, chainIntegrand next density g i n z q := by | |
| rw [parameterChain_eq_nested next density hfinite g i n z] | |
| exact integrateControls_eq n ((chainIntegrand_measurable next density hnext hdensity hg i n).comp | |
| measurable_prodMk_left) | |
| end BoltzmannNonuniqueness.FiveGainMap | |
| end | |
| end OAI | |