openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Couplings /OneCoordinateConditional.lean
| import OAI.Analysis.IrrationalRotation.Model.OneCoordinateConditional | |
| import OAI.Analysis.IrrationalRotation.Analysis.FiniteKernelMixture | |
| import OAI.Analysis.IrrationalRotation.Product.OneCoordinateDensity | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal ProbabilityTheory | |
| namespace Invariant233.OneCoordinateCoupling | |
| open CouplingMixture CouplingMarginalBounds FiniteKernelMixture | |
| private lemma finite_total (q : ℝ) (p : Fin 5 → ℝ) (u : Fin 5 → ℝ → ℝ) (x : ℝ) : | |
| ENNReal.ofReal q + ∑ i, ENNReal.ofReal (p i) * ENNReal.ofReal (u i x) ≠ ∞ := by | |
| apply ENNReal.add_ne_top.mpr | |
| refine ⟨ENNReal.ofReal_ne_top, ENNReal.sum_ne_top.mpr ?_⟩ | |
| intro i _ | |
| exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top | |
| section SourceParameters | |
| variable {κ m a b t : ℝ} | |
| (hκ0 : 0 ≤ κ) (hκ : κ ≤ 1 / 100) (hm : 100 ≤ m) | |
| (ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4) | |
| (hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4) | |
| (ht0 : 0 ≤ t) (ht : t ≤ 1) | |
| include hκ0 hκ hm ha ha' hb hb' ht0 ht | |
| lemma rightGivenLeft_isMarkov : IsMarkovKernel (rightGivenLeft κ m a b t) := by | |
| have hm0 : 0 < m := by linarith | |
| have : ∀ i, IsProbabilityMeasure (rightLaw m a b i) := | |
| rightLaw_isProbabilityMeasure (by linarith) (width_pos hm0 ha) ha' (width_pos hm0 hb) hb' | |
| apply diagonalReplacementKernel_isMarkov | |
| · apply ENNReal.ofReal_ne_zero_iff.mpr | |
| have := (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1 | |
| linarith | |
| · exact finite_total _ _ _ | |
| lemma leftGivenRight_isMarkov : IsMarkovKernel (leftGivenRight κ m a b t) := by | |
| have hm0 : 0 < m := by linarith | |
| have : ∀ i, IsProbabilityMeasure (leftLaw m a b i) := | |
| leftLaw_isProbabilityMeasure hm0 (width_pos hm0 ha) ha' (width_pos hm0 hb) hb' | |
| apply diagonalReplacementKernel_isMarkov | |
| · apply ENNReal.ofReal_ne_zero_iff.mpr | |
| have := (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1 | |
| linarith | |
| · exact finite_total _ _ _ | |
| lemma rightGivenLeft_disintegration : | |
| (pairLaw κ m a b t).fst ⊗ₘ rightGivenLeft κ m a b t = pairLaw κ m a b t := by | |
| have hm0 : 0 < m := by linarith | |
| have ha0 := width_pos hm0 ha | |
| have hb0 := width_pos hm0 hb | |
| have : ∀ i, IsProbabilityMeasure (rightLaw m a b i) := | |
| rightLaw_isProbabilityMeasure (by linarith) ha0 ha' hb0 hb' | |
| have := rightGivenLeft_isMarkov hκ0 hκ hm ha ha' hb hb' ht0 ht | |
| have hq : ENNReal.ofReal (diagonalMass κ m a b t) ≠ 0 := by | |
| apply ENNReal.ofReal_ne_zero_iff.mpr | |
| have := (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1 | |
| linarith | |
| let d : ℝ → ℝ≥0∞ := fun x => ENNReal.ofReal (diagonalMass κ m a b t) + | |
| ∑ i, ENNReal.ofReal (componentProbabilities κ m a b t i) * | |
| ENNReal.ofReal (leftDensity m a b i x) | |
| have hpair : (baseLaw.withDensity d) ⊗ₘ rightGivenLeft κ m a b t = pairLaw κ m a b t := by | |
| unfold rightGivenLeft d | |
| rw [diagonalReplacementKernel_disintegration _ _ hq] | |
| · unfold pairLaw mixture CouplingMixture.diagonal | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [leftLaw_eq_withDensity (by linarith) ha0 ha' hb0 hb'] | |
| · exact finite_total _ _ _ | |
| have hfst := congrArg Measure.fst hpair | |
| rw [Measure.fst_compProd] at hfst | |
| rw [← hfst] | |
| exact hpair | |
| lemma leftGivenRight_disintegration : | |
| (pairLaw κ m a b t).snd ⊗ₘ leftGivenRight κ m a b t = | |
| (pairLaw κ m a b t).map Prod.swap := by | |
| have hm0 : 0 < m := by linarith | |
| have ha0 := width_pos hm0 ha | |
| have hb0 := width_pos hm0 hb | |
| have : ∀ i, IsProbabilityMeasure (leftLaw m a b i) := | |
| leftLaw_isProbabilityMeasure hm0 ha0 ha' hb0 hb' | |
| have : ∀ i, IsProbabilityMeasure (rightLaw m a b i) := | |
| rightLaw_isProbabilityMeasure (by linarith) ha0 ha' hb0 hb' | |
| have := leftGivenRight_isMarkov hκ0 hκ hm ha ha' hb hb' ht0 ht | |
| have hq : ENNReal.ofReal (diagonalMass κ m a b t) ≠ 0 := by | |
| apply ENNReal.ofReal_ne_zero_iff.mpr | |
| have := (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1 | |
| linarith | |
| have hswap : (pairLaw κ m a b t).map Prod.swap = | |
| ENNReal.ofReal (diagonalMass κ m a b t) • CouplingMixture.diagonal baseLaw + | |
| ∑ i, ENNReal.ofReal (componentProbabilities κ m a b t i) • | |
| (rightLaw m a b i).prod (leftLaw m a b i) := by | |
| unfold pairLaw mixture | |
| rw [Measure.map_add _ _ measurable_swap, map_finsetSum _ _ measurable_swap] | |
| simp only [Measure.map_smul _ measurable_swap.aemeasurable, Measure.prod_swap] | |
| congr 1 | |
| unfold CouplingMixture.diagonal | |
| rw [Measure.map_map measurable_swap (show Measurable (fun x : ℝ => (x, x)) from | |
| measurable_id.prodMk measurable_id)] | |
| rfl | |
| let d : ℝ → ℝ≥0∞ := fun x => ENNReal.ofReal (diagonalMass κ m a b t) + | |
| ∑ i, ENNReal.ofReal (componentProbabilities κ m a b t i) * | |
| ENNReal.ofReal (rightDensity m a b i x) | |
| have hpair : (baseLaw.withDensity d) ⊗ₘ leftGivenRight κ m a b t = | |
| (pairLaw κ m a b t).map Prod.swap := by | |
| unfold leftGivenRight d | |
| rw [diagonalReplacementKernel_disintegration _ _ hq] | |
| · rw [hswap] | |
| unfold CouplingMixture.diagonal | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [rightLaw_eq_withDensity (by linarith) ha0 ha' hb0 hb'] | |
| · exact finite_total _ _ _ | |
| have hfst := congrArg Measure.fst hpair | |
| rw [Measure.fst_compProd, Measure.fst_map_swap] at hfst | |
| rw [← hfst] | |
| exact hpair | |
| end SourceParameters | |
| end Invariant233.OneCoordinateCoupling | |
| end OAI | |