| import OAI.Probability.ContinuumTransition.TailPressure |
|
|
| namespace OAI |
|
|
| section |
| open scoped Topology BigOperators |
| open Set Filter Metric |
| noncomputable section |
| namespace ContinuumTemperature |
|
|
| def ConstructionData.profile {p : ℝ} (C : ConstructionData p) : ℝ → ℝ := |
| physicalProfile C.t C.masses C.ranges |
|
|
| def ConstructionData.pressure {p : ℝ} (C : ConstructionData p) : Parameter → ℝ := |
| physicalPressure C.profile C.t |
|
|
| lemma ConstructionData.pressure_convex {p : ℝ} (C : ConstructionData p) : |
| ConvexOn ℝ pressureDomain C.pressure := |
| physical_convex_bounded C.cap.t_large C.masses_nonneg C.ranges_one C.masses_summable (C.cap.weak C.mass_bound) |
|
|
| lemma ConstructionData.pressure_tendsto {p : ℝ} (C : ConstructionData p) {x : Parameter} (hx : x ∈ pressureDomain) : |
| Tendsto (fun n => (C.stages n).pressure x) atTop (𝓝 (C.pressure x)) := by |
| have hb (n : ℕ) := physical_pressure_tail_bound C.cap.t_large hx |
| (prefixMass_nonneg C.masses_nonneg n) (tailMass_nonneg C.masses_nonneg n) C.ranges_one |
| (prefixMass_summable C.masses_nonneg C.masses_summable n) |
| (tailMass_summable C.masses_nonneg C.masses_summable n) |
| (by rw [prefixMass_add_tailMass]; exact C.cap.small C.mass_bound) x.2 |
| have hb' (n : ℕ) : 0 ≤ C.pressure x-(C.stages n).pressure x ∧ |
| C.pressure x-(C.stages n).pressure x ≤ |
| 108*2197*x.1*kernelHeight*momentConstant C.t x.1 (C.t*x.2)*(∑' k, tailMass C.masses n k) := by |
| have hh := hb n |
| rw [prefixMass_add_tailMass,← C.reference_eq_prefix n] at hh |
| exact hh |
| have hd : Tendsto (fun n => C.pressure x-(C.stages n).pressure x) atTop (𝓝 0) := by |
| apply squeeze_zero (fun n => (hb' n).1) (fun n => (hb' n).2) |
| simpa only [mul_zero] using (tailMass_tsum_tendsto C.masses_nonneg C.masses_summable).const_mul |
| (108*2197*x.1*kernelHeight*momentConstant C.t x.1 (C.t*x.2)) |
| simpa only [sub_sub_cancel,sub_zero] using (tendsto_const_nhds (x := C.pressure x)).sub hd |
|
|
| lemma initial_quarter_subset : ball initialPoint (1/4) ⊆ pressureDomain := |
| ball_subset_closedBall.trans ((closedBall_subset_closedBall (by norm_num : (1:ℝ)/4 ≤ 1/2)).trans |
| (initial_ball_moderate.trans moderateBox_subset)) |
|
|
| lemma ConstructionData.ideal_tendsto {p : ℝ} (C : ConstructionData p) {x : Parameter} |
| (hx : x ∈ ball initialPoint (1/4)) : |
| Tendsto (fun n => (C.stages n).ideal x) atTop (𝓝 (C.pressure x)) := by |
| have hd : Tendsto (fun n => (C.stages n).ideal x-(C.stages (n+1)).pressure x) atTop (𝓝 0) := by |
| apply squeeze_zero_norm _ geometricBudget_tendsto |
| intro n |
| simpa only [Real.norm_eq_abs,abs_sub_comm] using (C.transitions n).approximation x (ball_subset_closedBall hx) |
| have hn := (C.pressure_tendsto (initial_quarter_subset hx)).comp (tendsto_add_atTop_nat 1) |
| simpa only [Function.comp_def,sub_add_cancel,zero_add] using hd.add hn |
|
|
| lemma ConstructionData.ideal_locally_uniform {p : ℝ} (C : ConstructionData p) : |
| TendstoLocallyUniformlyOn (fun n => (C.stages n).ideal) C.pressure atTop (ball initialPoint (1/4)) := by |
| apply convex_pointwise_locally_uniform isOpen_ball |
| · exact Eventually.of_forall (fun n => ((C.stages n).convex C.cap).subset initial_quarter_subset (convex_ball _ _)) |
| · exact fun _ hx => C.ideal_tendsto hx |
|
|
| lemma ConstructionData.limit_support_ball {p : ℝ} (C : ConstructionData p) (hp : 0 < p) |
| {x : Parameter} (hx : Tendsto (fun n => (C.stages n).x) atTop (𝓝 x)) |
| (hnear : dist x initialPoint ≤ 1/8) : |
| closedBall (triangleCenter p) C.γ ⊆ subgradients pressureDomain C.pressure x := by |
| have hxW : x ∈ ball initialPoint (1/4) := hnear.trans_lt (by norm_num : (1:ℝ)/8 < 1/4) |
| have hc := C.pressure_convex.continuousOn isOpen_pressureDomain |
| have heval := C.ideal_locally_uniform.tendsto_comp (hc.mono initial_quarter_subset x hxW) hxW |
| (tendsto_nhdsWithin_iff.mpr ⟨hx,Eventually.of_forall (fun n => |
| (C.stages n).x_near.trans (by norm_num : (1:ℝ)/8 < 1/4))⟩) |
| intro v hv |
| apply supportOn_of_local C.pressure_convex (initial_quarter_subset hxW) |
| filter_upwards [isOpen_ball.mem_nhds hxW] with y hy |
| have hd : Tendsto (fun n => pDot v (y-(C.stages n).x)) atTop (𝓝 (pDot v (y-x))) := by |
| exact (show Continuous (fun z : Parameter => pDot v (y-z)) by unfold pDot; fun_prop).continuousAt.tendsto.comp hx |
| apply le_of_tendsto_of_tendsto (heval.add hd) (C.ideal_tendsto hy) |
| exact Eventually.of_forall (fun n => |
| ((C.stages n).triple.support_ball hp (groupRadius_le C.τ_pos.le n) C.τ_small C.geometry hv) y |
| (initial_quarter_subset hy)) |
| |
| end ContinuumTemperature |
| end |
| end |
| |
| end OAI |
| |
