openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Collision /KernelInvariantOrder.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.AeKernelHistoryOrder | |
| import Mathlib.Probability.Kernel.Composition.MeasureComp | |
| namespace OAI | |
| section | |
| /-! # Finite sums commute with applying a kernel to a measure -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped ProbabilityTheory | |
| namespace BoltzmannNonuniqueness | |
| variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] | |
| theorem kernel_finset_comp_measure {ι : Type*} (I : Finset ι) (K : ι → Kernel X Y) | |
| (μ : Measure X) : | |
| ((∑ i ∈ I, K i) : Kernel X Y) ∘ₘ μ = ∑ i ∈ I, K i ∘ₘ μ := by | |
| classical | |
| induction I using Finset.induction_on with | |
| | empty => simp | |
| | @insert i I hi ih => simp only [Finset.sum_insert hi,Measure.add_comp,ih] | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| section | |
| /-! # Finite history comparison only needs kernel order on the reachable region -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ProbabilityTheory | |
| namespace BoltzmannNonuniqueness | |
| variable {X : Type*} [MeasurableSpace X] | |
| theorem kernel_pow_le_on_invariant (K L : Kernel X X) (A : Set X) | |
| (hinv : ∀ x ∈ A, ∀ᵐ y ∂K x, y ∈ A) (hle : ∀ x ∈ A, K x ≤ L x) | |
| (n : ℕ) : ∀ x ∈ A, (K^n) x ≤ (L^n) x := by | |
| induction n with | |
| | zero => intro x hx; exact le_rfl | |
| | succ n ih => | |
| intro x hx | |
| rw [show K^(n+1) = (K^n) ∘ₖ K by simpa only [pow_one] using Kernel.pow_add K n 1, | |
| show L^(n+1) = (L^n) ∘ₖ L by simpa only [pow_one] using Kernel.pow_add L n 1] | |
| have h : ∀ᵐ y ∂K x, (K^n) y ≤ (L^n) y := by | |
| filter_upwards [hinv x hx] with y hy | |
| exact ih y hy | |
| exact KernelHistoryOrder.measure_comp_mono_ae h (hle x hx) | |
| end BoltzmannNonuniqueness | |
| end | |
| end | |
| end OAI | |