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caec230 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.MathematicalPhysics.RapidForcing.MachineDigits
namespace OAI
open scoped BigOperators ENNReal Topology
open Set MeasureTheory
namespace RapidForcing
namespace Machine
variable (M : Machine)
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]
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
lemma terminalDigit_le_one (d : ScaleData) (k : ℕ) : M.terminalDigit d k ≤ 1 := by
unfold terminalDigit
split_ifs <;> omega
end Machine
@[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}
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 _)
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]
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
end OAI
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