openAI-math-dataset / lean /OAI /Probability /LogConcave /JetEstimates /ClosedWireLintegralBound.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.JetEstimates.WireStep | |
| import OAI.Probability.LogConcave.TensorGraphs.Vertex | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped ENNReal NNReal Topology | |
| section UpperProof | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal RealInnerProductSpace Topology | |
| open Function MeasureTheory Set Filter | |
| open scoped Topology NNReal | |
| namespace LogConcaveSampling.TensorEnergy | |
| open MeasureTheory | |
| open scoped ENNReal NNReal | |
| theorem closed_wire_lintegral_bound | |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} | |
| {D I O S : ℕ → Type*} [∀k,Fintype (D k)] [∀k,Fintype (I k)] | |
| [∀k,Fintype (O k)] [∀k,Fintype (S k)] [∀k,DecidableEq (S k)] | |
| [DecidableEq (D 0)] | |
| (A : ∀k,Ω → O k → I k → ℝ) (C : ℕ → Ω → ℝ) (M : ℕ → ℝ≥0∞) | |
| (hC : ∀k x,0≤C k x) (hA : ∀k x,Bound (A k x) ((C k x)^2)) | |
| (ei : ∀k,D k ≃ I k×S k) (eo : ∀k,D (k+1) ≃ O k×S k) | |
| (n : ℕ) (hn : 0<n) (close : D 0 ≃ D n) | |
| (hm : ∀k<n,AEMeasurable (fun x => ENNReal.ofReal (C k x)) μ) | |
| (hb : ∀k<n,(∫⁻x,(ENNReal.ofReal (C k x))^n ∂μ)≤(M k)^n) : | |
| (∫⁻x,ENNReal.ofReal |∑i,circuit (fun k => wireStep (A k x) (ei k) (eo k)) n (close i) i| ∂μ)≤ | |
| (Fintype.card (D 0):ℝ≥0∞)*(∏k∈Finset.range n,M k) := by | |
| have hp (x : Ω) : | |
| ENNReal.ofReal |∑i,circuit (fun k => wireStep (A k x) (ei k) (eo k)) n (close i) i|≤ | |
| (Fintype.card (D 0):ℝ≥0∞)*(∏k∈Finset.range n,ENNReal.ofReal (C k x)) := by | |
| have hh := ENNReal.ofReal_le_ofReal (closed_wire_circuit_bound (fun k => A k x) | |
| (fun k => C k x) (fun k => hC k x) (fun k => hA k x) ei eo n close) | |
| rw [ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_natCast, | |
| ENNReal.ofReal_prod_of_nonneg (fun k _ => hC k x)] at hh | |
| exact hh | |
| calc | |
| _ ≤ ∫⁻x,(Fintype.card (D 0):ℝ≥0∞)*(∏k∈Finset.range n,ENNReal.ofReal (C k x)) ∂μ := | |
| lintegral_mono hp | |
| _ = (Fintype.card (D 0):ℝ≥0∞)*(∫⁻x,∏k∈Finset.range n,ENNReal.ofReal (C k x) ∂μ) := by | |
| exact lintegral_const_mul' _ _ (ENNReal.natCast_ne_top _) | |
| _ ≤ _ := by | |
| apply mul_le_mul' le_rfl | |
| apply NetworkMoments.lintegral_prod_le (Finset.range n) ⟨0,Finset.mem_range.mpr hn⟩ | |
| · exact fun k hk => hm k (Finset.mem_range.mp hk) | |
| · simpa only [Finset.card_range] using fun k hk => hb k (Finset.mem_range.mp hk) | |
| end LogConcaveSampling.TensorEnergy | |
| end UpperProof | |
| end | |
| end | |
| end | |
| end OAI | |