| import OAI.Probability.ContinuumTransition.ConstructionSequence |
|
|
| namespace OAI |
|
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| section |
| open scoped Topology BigOperators |
| open Set Filter Metric |
| noncomputable section |
| namespace ContinuumTemperature |
|
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| def ConstructionData.masses {p : ℝ} (C : ConstructionData p) (n : ℕ) : ℝ := (C.stages n).a |
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| def prefixMass (σ : ℕ → ℝ) (n k : ℕ) : ℝ := if k < n then σ k else 0 |
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| def tailMass (σ : ℕ → ℝ) (n k : ℕ) : ℝ := if n ≤ k then σ k else 0 |
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| lemma prefixMass_nonneg {σ : ℕ → ℝ} (hσ : ∀ k, 0 ≤ σ k) (n k : ℕ) : 0 ≤ prefixMass σ n k := by |
| unfold prefixMass |
| split_ifs |
| · exact hσ k |
| · exact le_refl _ |
|
|
| lemma tailMass_nonneg {σ : ℕ → ℝ} (hσ : ∀ k, 0 ≤ σ k) (n k : ℕ) : 0 ≤ tailMass σ n k := by |
| unfold tailMass |
| split_ifs |
| · exact hσ k |
| · exact le_refl _ |
|
|
| lemma prefixMass_add_tailMass (σ : ℕ → ℝ) (n : ℕ) : |
| (fun k => prefixMass σ n k+tailMass σ n k) = σ := by |
| funext k |
| by_cases hk : k < n |
| · simp [prefixMass,tailMass,hk,Nat.not_le.mpr hk] |
| · simp [prefixMass,tailMass,hk,Nat.le_of_not_gt hk] |
|
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| lemma prefixMass_summable {σ : ℕ → ℝ} (hσ : ∀ k, 0 ≤ σ k) (hs : Summable σ) (n : ℕ) : |
| Summable (prefixMass σ n) := by |
| apply hs.of_nonneg_of_le (prefixMass_nonneg hσ n) |
| intro k |
| unfold prefixMass |
| split_ifs <;> simp_all |
|
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| lemma tailMass_summable {σ : ℕ → ℝ} (hσ : ∀ k, 0 ≤ σ k) (hs : Summable σ) (n : ℕ) : |
| Summable (tailMass σ n) := by |
| apply hs.of_nonneg_of_le (tailMass_nonneg hσ n) |
| intro k |
| unfold tailMass |
| split_ifs <;> simp_all |
|
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| lemma tsum_prefixMass (σ : ℕ → ℝ) (n : ℕ) : |
| ∑' k, prefixMass σ n k = ∑ k ∈ Finset.range n, σ k := by |
| rw [tsum_eq_sum (s := Finset.range n) (fun k hk => by |
| simp only [Finset.mem_range] at hk |
| simp only [prefixMass,ite_eq_right hk])] |
| apply Finset.sum_congr rfl |
| intro k hk |
| simp only [prefixMass,Finset.mem_range.mp hk,ite_true] |
|
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| lemma tailMass_tsum_tendsto {σ : ℕ → ℝ} (hσ : ∀ k, 0 ≤ σ k) (hs : Summable σ) : |
| Tendsto (fun n => ∑' k, tailMass σ n k) atTop (𝓝 0) := by |
| have he (n : ℕ) : ∑' k, tailMass σ n k = (∑' k, σ k)-∑ k ∈ Finset.range n, σ k := by |
| have hh := (prefixMass_summable hσ hs n).tsum_add (tailMass_summable hσ hs n) |
| rw [prefixMass_add_tailMass,tsum_prefixMass] at hh |
| linarith |
| simp_rw [he] |
| simpa only [sub_self] using (tendsto_const_nhds (x := ∑' k, σ k)).sub hs.hasSum.tendsto_sum_nat |
|
|
| lemma ConstructionData.reference_mass {p : ℝ} (C : ConstructionData p) (n : ℕ) : |
| ∑' k, (C.stages n).σ k = ∑ k ∈ Finset.range n, C.masses k := by |
| induction n with |
| | zero => simp [C.initial_zero] |
| | succ n ih => |
| rw [(C.transitions n).mass_eq,tsum_prepend _ (C.stages n).σ_summable,ih,Finset.sum_range_succ] |
| dsimp [ConstructionData.masses] |
| ring |
|
|
| lemma ConstructionData.masses_nonneg {p : ℝ} (C : ConstructionData p) (n : ℕ) : 0 ≤ C.masses n := |
| (C.stages n).a_pos.le |
|
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| lemma ConstructionData.sum_masses_le {p : ℝ} (C : ConstructionData p) (n : ℕ) : |
| ∑ k ∈ Finset.range n, C.masses k ≤ firstMass p+1 := by |
| rw [← C.reference_mass] |
| exact (C.stages n).base_mass |
|
|
| lemma ConstructionData.masses_summable {p : ℝ} (C : ConstructionData p) : Summable C.masses := |
| summable_of_sum_range_le C.masses_nonneg C.sum_masses_le |
|
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| lemma ConstructionData.mass_bound {p : ℝ} (C : ConstructionData p) : ∑' k, C.masses k ≤ firstMass p+1 := |
| Real.tsum_le_of_sum_range_le C.masses_nonneg C.sum_masses_le |
|
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| lemma ConstructionData.ranges_one {p : ℝ} (C : ConstructionData p) (n : ℕ) : 1 ≤ C.ranges n := by |
| have hh := (C.transitions n).range_large |
| linarith |
|
|
| lemma ConstructionData.reference_profile {p : ℝ} (C : ConstructionData p) (n : ℕ) (r : ℝ) : |
| physicalProfile C.t (C.stages n).σ (C.stages n).R r = |
| coreProfile C.t r-C.t*∑ k ∈ Finset.range n, C.masses k*scaledKernelProfile (C.ranges k) r := by |
| induction n with |
| | zero => simp [C.initial_zero,physicalProfile_zero] |
| | succ n ih => |
| rw [(C.transitions n).mass_eq,(C.transitions n).range_eq, |
| physicalProfile_prepend C.t (C.stages n).σ_nonneg (C.stages n).R_one (C.stages n).σ_summable,ih, |
| Finset.sum_range_succ] |
| dsimp [ConstructionData.masses] |
| ring |
|
|
| lemma attraction_prefixMass (t : ℝ) (σ R : ℕ → ℝ) (n : ℕ) (r : ℝ) : |
| attraction t (prefixMass σ n) R r = t*∑ k ∈ Finset.range n, σ k*scaledKernelProfile (R k) r := by |
| unfold attraction |
| congr 1 |
| rw [tsum_eq_sum (s := Finset.range n) (fun k hk => by |
| simp only [Finset.mem_range] at hk |
| simp only [prefixMass,ite_eq_right hk,zero_mul])] |
| apply Finset.sum_congr rfl |
| intro k hk |
| simp only [prefixMass,Finset.mem_range.mp hk,ite_true] |
|
|
| lemma ConstructionData.reference_eq_prefix {p : ℝ} (C : ConstructionData p) (n : ℕ) : |
| physicalProfile C.t (C.stages n).σ (C.stages n).R = physicalProfile C.t (prefixMass C.masses n) C.ranges := by |
| funext r |
| rw [C.reference_profile] |
| unfold physicalProfile |
| rw [attraction_prefixMass] |
|
|
| end ContinuumTemperature |
| end |
| end |
|
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| end OAI |
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|
