| import OAI.MathematicalPhysics.RapidForcing.MachineDigits |
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| namespace OAI |
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| open scoped BigOperators ENNReal Topology |
| open Set MeasureTheory |
| namespace RapidForcing |
| namespace Machine |
| variable (M : Machine) |
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|
| lemma initialDigit_eq (w : M.Input) : |
| M.initialDigit w = M.initial.val + (M.scaleData w).S * |
| ((M.scaleData w).P 0 * Nat.ofDigits (M.scaleData w).A (w.map Fin.val)) := by |
| simp only [initialDigit, Nat.ofDigits_eq_sum_mapIdx, List.mapIdx_eq_zipIdx_map] |
|
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| lemma initialDigit_lt (w : M.Input) : M.initialDigit w < (M.scaleData w).capacity 0 := by |
| let d := M.scaleData w |
| have hA : 1 < d.A := lt_of_le_of_lt M.symbols_pos (M.alphabet_lt_A w) |
| have hw : ∀ z ∈ w.map Fin.val, z < d.A := by |
| intro z hz |
| obtain ⟨a, _, rfl⟩ := List.mem_map.mp hz |
| exact lt_trans a.isLt (M.alphabet_lt_A w) |
| have hT := Nat.ofDigits_lt_base_pow_length hA hw |
| simp only [List.length_map] at hT |
| have hT' : Nat.ofDigits d.A (w.map Fin.val) < d.A ^ (2 * 0 + d.B + 1) := by |
| apply lt_of_lt_of_le hT |
| apply Nat.pow_le_pow_right d.one_le_A |
| change w.length ≤ 2 * 0 + w.length + 1 |
| omega |
| rw [M.initialDigit_eq w, d.capacity_eq] |
| have hq : M.initial.val < d.S := lt_trans M.initial.isLt (M.states_lt_S w) |
| have hj : 0 < d.P 0 := d.P_pos _ |
| have h := radix_pair_lt hq (radix_pair_lt hj hT') |
| simpa only [zero_add, Nat.mul_assoc] using h |
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| lemma terminalDigit_le_one (d : ScaleData) (k : ℕ) : M.terminalDigit d k ≤ 1 := by |
| unfold terminalDigit |
| split_ifs <;> omega |
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|
| end Machine |
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| @[simp] lemma vec_zero_coord (a b c : ℝ) : vec a b c 0 = a := rfl |
| @[simp] lemma vec_one_coord (a b c : ℝ) : vec a b c 1 = b := rfl |
| @[simp] lemma vec_two_coord (a b c : ℝ) : vec a b c 2 = c := rfl |
| @[simp] lemma vec_all_zero : vec 0 0 0 = (0 : Space) := by |
| ext i |
| fin_cases i <;> rfl |
|
|
| lemma theta_nonneg (t : ℝ) : 0 ≤ θ t := Real.smoothTransition.nonneg _ |
| lemma theta_le_one (t : ℝ) : θ t ≤ 1 := Real.smoothTransition.le_one _ |
| lemma theta_zero {t : ℝ} (h : t ≤ 1 / 4) : θ t = 0 := by |
| exact Real.smoothTransition.zero_of_nonpos (by linarith) |
| lemma theta_one {t : ℝ} (h : 3 / 4 ≤ t) : θ t = 1 := by |
| exact Real.smoothTransition.one_of_one_le (by linarith) |
| lemma theta_smooth : ContDiff ℝ (⊤ : ℕ∞) θ := by |
| unfold θ |
| fun_prop |
|
|
| def supportBox (c : Space) (r : ℝ) : Set Space := {x | ∀ i : Fin 3, |x i - c i| ≤ r} |
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| lemma supportBox_closed (c : Space) (r : ℝ) : IsClosed (supportBox c r) := by |
| unfold supportBox |
| simp only [ofPred_forall] |
| apply isClosed_iInter |
| intro i |
| exact isClosed_le (((EuclideanSpace.proj i).continuous.sub continuous_const).abs) continuous_const |
|
|
| lemma zeta_nonneg (v : Space) : 0 ≤ ζ v := by |
| apply Finset.prod_nonneg |
| intro i _ |
| exact mul_nonneg (Real.smoothTransition.nonneg _) (Real.smoothTransition.nonneg _) |
|
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| lemma zeta_le_one (v : Space) : ζ v ≤ 1 := by |
| apply Finset.prod_le_one₀ |
| · intro i _ |
| exact mul_nonneg (Real.smoothTransition.nonneg _) (Real.smoothTransition.nonneg _) |
| · intro i _ |
| exact (mul_le_of_le_one_left (Real.smoothTransition.nonneg _) |
| (Real.smoothTransition.le_one _)).trans (Real.smoothTransition.le_one _) |
|
|
| lemma zeta_zero {v : Space} {i : Fin 3} (h : 1 / 8 ≤ |v i|) : ζ v = 0 := by |
| unfold ζ |
| apply Finset.prod_eq_zero (Finset.mem_univ i) |
| rcases le_or_gt (v i) 0 with hsign | hsign |
| · rw [abs_of_nonpos hsign] at h |
| rw [Real.smoothTransition.zero_of_nonpos (by linarith : 16 * (v i + 1 / 8) ≤ 0), |
| zero_mul] |
| · rw [abs_of_nonneg hsign.le] at h |
| rw [Real.smoothTransition.zero_of_nonpos (by linarith : 16 * (1 / 8 - v i) ≤ 0), |
| mul_zero] |
|
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| lemma zeta_one {v : Space} (h : ∀ i : Fin 3, |v i| ≤ 1 / 16) : ζ v = 1 := by |
| apply Finset.prod_eq_one |
| intro i _ |
| obtain ⟨h₁, h₂⟩ := abs_le.mp (h i) |
| rw [Real.smoothTransition.one_of_one_le (by linarith : 1 ≤ 16 * (v i + 1 / 8)), |
| Real.smoothTransition.one_of_one_le (by linarith : 1 ≤ 16 * (1 / 8 - v i)), one_mul] |
|
|
| lemma zeta_smooth : ContDiff ℝ (⊤ : ℕ∞) ζ := by |
| unfold ζ |
| fun_prop |
|
|
| lemma curl_tsupport_subset (v : Space → Space) : tsupport (curl v) ⊆ tsupport v := by |
| apply closure_minimal _ (isClosed_tsupport v) |
| intro x hx |
| by_contra hn |
| have hd := fderiv_of_notMem_tsupport ℝ hn |
| apply hx |
| simp [curl, hd] |
|
|
| lemma movingCurl_tsupport_subset (c : ℝ → Space) {δ : ℝ} (hδ : 0 < δ) (σ : ℝ) : |
| tsupport (movingCurl c δ σ) ⊆ supportBox (c σ) (δ / 8) := by |
| apply (curl_tsupport_subset _).trans |
| apply closure_minimal _ (supportBox_closed _ _) |
| intro x hx |
| change ζ (δ⁻¹ • (x - c σ)) • ((1 / 2 : ℝ) • cross (deriv c σ) (x - c σ)) ≠ 0 at hx |
| intro i |
| by_contra hn |
| have hi : δ / 8 < |x i - c σ i| := lt_of_not_ge hn |
| have hco : (δ⁻¹ • (x - c σ)) i = (x i - c σ i) / δ := by |
| simp [div_eq_mul_inv, mul_comm] |
| have hz : ζ (δ⁻¹ • (x - c σ)) = 0 := by |
| apply zeta_zero (i := i) |
| rw [hco, abs_div, abs_of_pos hδ] |
| apply (le_div_iff₀ hδ).mpr |
| linarith |
| exact hx (by rw [hz, zero_smul]) |
|
|
| end RapidForcing |
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| end OAI |
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