openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /VelocityDetection /BurstTotalReferenceSum.lean
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.BurstTotal | |
| import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.BurstBlockRawReferenceLaplacian | |
| namespace OAI | |
| noncomputable section | |
| namespace VelocityDetection.BurstTotal | |
| open Set Function Filter MeasureTheory | |
| open scoped Topology ContDiff BigOperators | |
| open ChartRouting BurstSchedule Periodization SpatialCalculus | |
| variable (A : RoutingData) (ν : ℝ) (tr : RoutingArray.Trace A) | |
| def referenceSum (L : ℕ) : ScalarField 2 := fun t X => | |
| ∑ k ∈ Finset.range L, BurstBlock.reference A ν tr k t X | |
| def barrierSum (L : ℕ) (t : ℝ) : ℝ := ∑ k ∈ Finset.range L, barrier A ν k t | |
| theorem reference_rest (hν : 0 ≤ ν) (k : ℕ) {t : ℝ} (ht : t ≤ 1) : | |
| BurstBlock.reference A ν tr k t = 0 := | |
| BurstBlock.reference_zero A ν tr k | |
| (Or.inl (localTime_before A ν hν k (ht.trans (by linarith [Nat.cast_nonneg (α := ℝ) k])))) | |
| theorem reference_transport (hν : 0 ≤ ν) (k : ℕ) | |
| (h : ∀ n < k+1, tr.stopped n = false) (t : ℝ) (X : Coord 2) : | |
| deriv (fun s => BurstBlock.reference A ν tr k s X) t + | |
| advection (field A ν) (BurstBlock.reference A ν tr k) t X = | |
| BurstBlock.source A ν (tr.address 0) k t X := by | |
| have ha : advection (field A ν) (BurstBlock.reference A ν tr k) t X = | |
| advection (BurstBlock.field A ν k) (BurstBlock.reference A ν tr k) t X := by | |
| by_cases ht : t ∈ Icc ((k:ℝ)+1) ((k:ℝ)+2) | |
| · simp only [advection,congrFun (field_at_block A ν hν k ht) X] | |
| · have hz : BurstBlock.reference A ν tr k t = 0 := by | |
| apply BurstBlock.reference_zero A ν tr k | |
| rcases lt_or_ge t ((k:ℝ)+1) with hlt | hge | |
| · exact Or.inl (localTime_before A ν hν k hlt.le) | |
| · exact Or.inr (localTime_after A ν hν k (le_of_lt (lt_of_not_ge (fun hle => ht ⟨hge,hle⟩)))) | |
| rw [congrFun (TimeBlocks.advection_zero (field A ν) hz) X, | |
| congrFun (TimeBlocks.advection_zero (BurstBlock.field A ν k) hz) X] | |
| rw [ha] | |
| exact BurstBlock.reference_transport A ν tr k h t X | |
| @[fun_prop] theorem contDiff_referenceSum (L : ℕ) | |
| (h : ∀ n < L, tr.stopped n = false) : | |
| ContDiff ℝ ∞ (uncurry (referenceSum A ν tr L)) := | |
| ContDiff.sum (fun k hk => BurstBlock.contDiff_reference A ν tr k | |
| (fun n hn => h n (lt_of_lt_of_le hn (Finset.mem_range.mp hk)))) | |
| theorem referenceSum_periodic (L : ℕ) (t : ℝ) : | |
| FactorsThrough (referenceSum A ν tr L t) PeriodicSpace.cover := by | |
| intro X Y hXY | |
| exact Finset.sum_congr rfl (fun k _ => BurstBlock.reference_periodic A ν tr k t hXY) | |
| theorem referenceSum_transport (hν : 0 ≤ ν) (L : ℕ) | |
| (h : ∀ n < L, tr.stopped n = false) {t : ℝ} (ht : t ≤ (L:ℝ)+1) (X : Coord 2) : | |
| deriv (fun s => referenceSum A ν tr L s X) t + | |
| advection (field A ν) (referenceSum A ν tr L) t X = source A ν (tr.address 0) t X := by | |
| have hsm (k : ℕ) (hk : k ∈ Finset.range L) := BurstBlock.contDiff_reference A ν tr k | |
| (fun n hn => h n (lt_of_lt_of_le hn (Finset.mem_range.mp hk))) | |
| change deriv (fun s => ∑ k ∈ Finset.range L, BurstBlock.reference A ν tr k s X) t + | |
| advection (field A ν) (fun s Y => ∑ k ∈ Finset.range L, BurstBlock.reference A ν tr k s Y) t X = _ | |
| have hd (k : ℕ) (hk : k ∈ Finset.range L) : | |
| DifferentiableAt ℝ (fun s => BurstBlock.reference A ν tr k s X) t := | |
| ((hsm k hk).comp (contDiff_id.prodMk contDiff_const)).differentiable (by simp) t | |
| rw [deriv_fun_sum hd, | |
| TimeBlocks.advection_sum (field A ν) (Finset.range L) (BurstBlock.reference A ν tr) | |
| (fun k hk s => ((hsm k hk).comp (contDiff_const.prodMk contDiff_id)).differentiable (by simp)), | |
| ← Finset.sum_add_distrib] | |
| have he : source A ν (tr.address 0) t X = | |
| ∑ k ∈ Finset.range L, BurstBlock.source A ν (tr.address 0) k t X := | |
| congrFun (TimeBlocks.eq_finite (sourceFamily A ν hν (tr.address 0)) ht) X | |
| rw [he] | |
| exact Finset.sum_congr rfl (fun k hk => reference_transport A ν tr hν k | |
| (fun n hn => h n (lt_of_lt_of_le hn (Finset.mem_range.mp hk))) t X) | |
| @[fun_prop] theorem contDiff_barrierSum (L : ℕ) : ContDiff ℝ 2 (barrierSum A ν L) := | |
| ContDiff.sum (fun k _ => contDiff_barrier A ν k) | |
| theorem deriv_barrierSum (L : ℕ) (t : ℝ) : deriv (barrierSum A ν L) t = | |
| ∑ k ∈ Finset.range L, deriv (barrier A ν k) t := | |
| deriv_fun_sum (fun k _ => (contDiff_barrier A ν k).differentiable (by norm_num) t) | |
| theorem barrierSum_bounds (L : ℕ) (t : ℝ) : barrierSum A ν L t ∈ Icc (0:ℝ) (1/8) := | |
| ⟨Finset.sum_nonneg (fun k _ => (barrier_bounds A ν k t).1), | |
| (Finset.sum_le_sum (fun k _ => (barrier_bounds A ν k t).2)).trans (error_sum_le L)⟩ | |
| theorem referenceSum_laplacian (hν : 0 ≤ ν) (L : ℕ) | |
| (h : ∀ n < L, tr.stopped n = false) (t : ℝ) (X : Coord 2) : | |
| ν*|laplacian (referenceSum A ν tr L) t X| ≤ deriv (barrierSum A ν L) t := by | |
| change ν*|laplacian (fun s Y => ∑ k ∈ Finset.range L, BurstBlock.reference A ν tr k s Y) t X| ≤ _ | |
| rw [TimeBlocks.laplacian_sum (Finset.range L) (BurstBlock.reference A ν tr) | |
| (fun k hk => BurstBlock.contDiff_reference A ν tr k | |
| (fun n hn => h n (lt_of_lt_of_le hn (Finset.mem_range.mp hk)))),deriv_barrierSum] | |
| apply (mul_le_mul_of_nonneg_left (Finset.abs_sum_le_sum_abs _ _) hν).trans | |
| rw [Finset.mul_sum] | |
| exact Finset.sum_le_sum (fun k hk => BurstBlock.reference_laplacian_barrier A ν tr hν k | |
| (fun n hn => h n (lt_of_lt_of_le hn (Finset.mem_range.mp hk))) t X) | |
| theorem scalar_prefix_error (hν : 0 < ν) (L : ℕ) | |
| (h : ∀ n < L, tr.stopped n = false) {t : ℝ} (ht : t ∈ Icc (0:ℝ) ((L:ℝ)+1)) | |
| (X : Coord 2) : |scalar A ν hν (tr.address 0) t X-referenceSum A ν tr L t X| ≤ 1/8 := by | |
| have hzero : referenceSum A ν tr L 0 X = 0 := by | |
| exact Finset.sum_eq_zero (fun k _ => congrFun (reference_rest A ν tr hν.le k (by norm_num)) X) | |
| have hh := PeriodicComparison.diffusion_error (g := source A ν (tr.address 0)) | |
| hν.le (by positivity : (0:ℝ) ≤ (L:ℝ)+1) | |
| ((contDiff_scalar A ν hν (tr.address 0)).of_le (WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top))) | |
| ((contDiff_referenceSum A ν tr L h).of_le (WithTop.coe_le_coe.mpr (show (2 : ℕ∞) ≤ ⊤ from le_top))) | |
| (scalar_periodic A ν hν (tr.address 0)) (referenceSum_periodic A ν tr L) | |
| (contDiff_barrierSum A ν L) | |
| (fun _ hs Y => scalar_residual A ν hν (tr.address 0) hs.1 Y) | |
| (fun _ hs Y => referenceSum_transport A ν tr hν.le L h hs.2 Y) | |
| (fun s _ Y => referenceSum_laplacian A ν tr hν.le L h s Y) | |
| (fun Y => by | |
| rw [scalar_initial,show referenceSum A ν tr L 0 Y = 0 from | |
| Finset.sum_eq_zero (fun k _ => congrFun (reference_rest A ν tr hν.le k (by norm_num)) Y), | |
| sub_self,abs_zero] | |
| exact (barrierSum_bounds A ν L 0).1) | |
| exact (hh t ht X).trans (barrierSum_bounds A ν L t).2 | |
| def velocity (hν : 0 < ν) (q : ℕ) : VectorField 3 := | |
| liftVelocity (field A ν) (scalar A ν hν q) | |
| theorem strip_representative (X : Coord 2) (hY : X 1 ∈ Icc (0:ℝ) 1) : | |
| ∃ Y : Coord 2, (∀ i, Y i ∈ Icc (0:ℝ) 1) ∧ Y 1 = X 1 ∧ | |
| PeriodicSpace.cover X = PeriodicSpace.cover Y := by | |
| let Y : Coord 2 := ![Int.fract (X 0),X 1] | |
| refine ⟨Y,?_,rfl,?_⟩ | |
| · intro i | |
| fin_cases i | |
| · exact ⟨Int.fract_nonneg _,(Int.fract_lt_one _).le⟩ | |
| · exact hY | |
| · apply cover_eq_iff.mpr | |
| refine ⟨![Int.floor (X 0),0],?_⟩ | |
| ext i | |
| fin_cases i | |
| · change X 0 = Int.fract (X 0)+(Int.floor (X 0):ℝ) | |
| exact (sub_add_cancel _ _).symm | |
| · change X 1 = X 1+(0:ℤ) | |
| simp | |
| theorem nonterminal_small (hν : 0 < ν) (hnever : ¬ tr.Stops) | |
| {t : ℝ} (ht : 0 ≤ t) (X : Coord 2) (hY : X 1 ∈ Ioo (1/32:ℝ) (1/8)) : | |
| scalar A ν hν (tr.address 0) t X ≤ 1/8 := by | |
| have hactive (n : ℕ) : tr.stopped n = false := by | |
| cases h : tr.stopped n | |
| · rfl | |
| · exact False.elim (hnever ⟨n,h⟩) | |
| obtain ⟨Y,hYcube,hYY,hXY⟩ := strip_representative X ⟨by linarith [hY.1],by linarith [hY.2]⟩ | |
| rw [scalar_periodic A ν hν (tr.address 0) t hXY] | |
| obtain ⟨L,hL⟩ := exists_nat_gt t | |
| have he := scalar_prefix_error A ν tr hν L (fun n _ => hactive n) ⟨ht,by linarith⟩ Y | |
| have hz : referenceSum A ν tr L t Y = 0 := by | |
| exact Finset.sum_eq_zero (fun k _ => BurstBlock.reference_zero_detector A ν tr k | |
| (fun n _ => hactive n) t hYcube (by rw [hYY]; exact hY.2)) | |
| rw [hz,sub_zero] at he | |
| exact (le_abs_self _).trans he | |
| theorem terminal_large (hν : 0 < ν) (hinit : tr.stopped 0 = false) | |
| (hreach : tr.Stops) : | |
| ∃ t : ℝ, 0 ≤ t ∧ ∃ X : Coord 2, X 1 ∈ Ioo (1/32:ℝ) (1/8) ∧ | |
| (3/4:ℝ) < scalar A ν hν (tr.address 0) t X := by | |
| let hex : ∃ n, tr.stopped n = true := hreach | |
| have hzero : Nat.find hex ≠ 0 := by | |
| intro hz | |
| have h := Nat.find_spec hex | |
| rw [hz,hinit] at h | |
| cases h | |
| obtain ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero hzero | |
| have hfirst : tr.stopped (k+1) = true := by simpa only [hk] using Nat.find_spec hex | |
| have hactive (n : ℕ) (hn : n < k+1) : tr.stopped n = false := | |
| Bool.eq_false_iff.mpr (Nat.find_min hex (by omega)) | |
| let t : ℝ := (k:ℝ)+1+duration A ν k/2 | |
| have hdur := duration_pos A ν hν.le k | |
| have hdurs := duration_small A ν hν.le k | |
| have ht : 0 ≤ t := by dsimp [t]; positivity | |
| have htt : t ∈ Icc ((k:ℝ)+1) ((k:ℝ)+2) := by dsimp [t]; constructor <;> linarith | |
| have hlocal : localTime A ν k t = 1/2 := by | |
| unfold localTime | |
| dsimp [t] | |
| field_simp | |
| ring | |
| have hclock : clock A ν k t = k+1 := by | |
| change phase (k+1) (localTime A ν k t) = _ | |
| rw [hlocal,phase_middle (k+1) (by norm_num : (1/2:ℝ) ∈ Icc (1/4:ℝ) (5/8))] | |
| push_cast | |
| rfl | |
| have hweight : weight A ν k t = 1 := by | |
| change amplitude (localTime A ν k t) = _ | |
| rw [hlocal] | |
| exact amplitude_middle (by norm_num) | |
| let X := BurstBlock.center A ν tr k t | |
| have hX : X 1 = 1/16 := by | |
| change ChartTrace.route A tr (k+1) (clock A ν k t) 1 = _ | |
| rw [hclock,ChartTrace.route_after A tr (k+1) (by simp)] | |
| simp [ChartTrace.vertex,ChartTrace.height,hfirst,origin] | |
| norm_num | |
| refine ⟨t,ht,X,by rw [hX]; norm_num,?_⟩ | |
| have he := scalar_prefix_error A ν tr hν (k+1) hactive | |
| (t := t) ⟨ht,by push_cast; linarith [htt.2]⟩ X | |
| have href : referenceSum A ν tr (k+1) t X = 1 := by | |
| change (∑ j ∈ Finset.range (k+1), BurstBlock.reference A ν tr j t X) = _ | |
| rw [Finset.sum_eq_single k] | |
| · exact (BurstBlock.reference_center A ν tr k hactive t).trans hweight | |
| · intro j hj hjk | |
| have hh : (j:ℝ)+2 ≤ t := by | |
| have hle : (j:ℝ)+1 ≤ k := by exact_mod_cast (show j < k by have := Finset.mem_range.mp hj; omega) | |
| linarith [htt.1] | |
| exact congrFun (BurstBlock.reference_zero A ν tr j | |
| (Or.inr (localTime_after A ν hν.le j hh))) X | |
| · exact fun h => (h (Finset.mem_range.mpr (Nat.lt_succ_self _))).elim | |
| rw [href] at he | |
| linarith [(abs_le.mp he).1] | |
| theorem detection (hν : 0 < ν) (hinit : tr.stopped 0 = false) : | |
| Observation.torusEvent (velocity A ν hν (tr.address 0)) ↔ tr.Stops := by | |
| constructor | |
| · rintro ⟨t,ht,x,hy₀,hy₁,hbig⟩ | |
| by_contra hn | |
| have hsmall := nonterminal_small A ν tr hν hn ht (horizontal x) ⟨hy₀,hy₁⟩ | |
| change (1/2:ℝ) < scalar A ν hν (tr.address 0) t (horizontal x) at hbig | |
| linarith | |
| · intro h | |
| obtain ⟨t,ht,X,hX,hbig⟩ := terminal_large A ν tr hν hinit h | |
| refine ⟨t,ht,Observation.join X 0,hX.1,hX.2,?_⟩ | |
| change (1/2:ℝ) < scalar A ν hν (tr.address 0) t (horizontal (Observation.join X 0)) | |
| rw [Observation.horizontal_join] | |
| linarith | |
| end VelocityDetection.BurstTotal | |
| end | |
| end OAI | |