| import OAI.MathematicalPhysics.AlternatingFlow.Model |
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| namespace OAI |
|
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| open scoped BigOperators ENNReal NNReal Topology ContDiff |
| open MeasureTheory |
| namespace AlternatingNS |
| namespace Scales |
|
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| def K (N n : ℕ) : ℕ := N + 2 * n + 2 |
|
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| def s (N n : ℕ) : ℕ := (N + 1) * 2 ^ ((n + 3) ^ 2) |
|
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| noncomputable def ε (b N n : ℕ) : ℝ := ((b : ℝ) ^ s N n)⁻¹ |
|
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| lemma exponent_succ (n : ℕ) : (n + 1 + 3) ^ 2 = (n + 3) ^ 2 + (2 * n + 7) := by |
| ring |
|
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| lemma s_succ (N n : ℕ) : s N (n + 1) = 2 ^ (2 * n + 7) * s N n := by |
| simp only [s, exponent_succ, pow_add] |
| ring |
|
|
| lemma pow_lower (n : ℕ) : 8 * (n + 1) ≤ 2 ^ ((n + 3) ^ 2) := by |
| have h : n + 1 ≤ 2 ^ n := Nat.lt_two_pow_self |
| have he : n + 3 ≤ (n + 3) ^ 2 := by nlinarith |
| calc |
| 8 * (n + 1) ≤ 8 * 2 ^ n := Nat.mul_le_mul_left 8 h |
| _ = 2 ^ (n + 3) := by simp [pow_add]; ring |
| _ ≤ 2 ^ ((n + 3) ^ 2) := Nat.pow_le_pow_right (by decide) he |
| |
| lemma s_lower (N n : ℕ) : 8 * (N + 1) * (n + 1) ≤ s N n := by |
| have h := Nat.mul_le_mul_left (N + 1) (pow_lower n) |
| simpa only [s, mul_left_comm, mul_assoc] using h |
| |
| lemma capacity_lt (N n : ℕ) : 1 + 2 * K N n < s N n := by |
| have h := s_lower N n |
| dsimp [K] |
| nlinarith |
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| lemma K_add_one_lt (N n : ℕ) : K N n + 1 < s N n := by |
| have := capacity_lt N n |
| omega |
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| lemma s_pos (N n : ℕ) : 0 < s N n := by |
| unfold s |
| positivity |
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|
| lemma strictMono_s (N : ℕ) : StrictMono (s N) := by |
| apply strictMono_nat_of_lt_succ |
| intro n |
| rw [s_succ] |
| have hp : 1 < 2 ^ (2 * n + 7) := one_lt_pow₀ (by decide) (by omega) |
| nlinarith [s_pos N n] |
|
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| lemma scale_gap (N n : ℕ) : s N n + 1 + 2 * K N n < s N (n + 1) := by |
| rw [s_succ] |
| have hc := capacity_lt N n |
| have hp : 2 ≤ 2 ^ (2 * n + 7) := by |
| exact Nat.le_pow (by omega) |
| nlinarith [s_pos N n] |
|
|
| lemma old_scale_gap (N j n : ℕ) (hjn : j < n) : |
| s N j + (1 + 2 * K N j) ≤ s N n := by |
| have h₁ := scale_gap N j |
| have h₂ := (strictMono_s N).monotone (Nat.succ_le_of_lt hjn) |
| simp only [Nat.succ_eq_add_one] at h₂ |
| omega |
| |
| lemma derivative_gap (N r n : ℕ) (hn : r ≤ n) : |
| s N n + (r + 1) * (s N n + 1 + K N n) ≤ s N (n + 1) := by |
| have hk := K_add_one_lt N n |
| have hp : 2 * r + 3 ≤ 2 ^ (2 * n + 7) := by |
| have ht : 2 * n + 7 < 2 ^ (2 * n + 7) := Nat.lt_two_pow_self |
| omega |
| rw [s_succ] |
| nlinarith |
|
|
| lemma s_zero_add (N n : ℕ) : s N 0 + n ≤ s N n := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| have h := strictMono_s N (Nat.lt_succ_self n) |
| simp only [Nat.succ_eq_add_one] at * |
| omega |
| |
| lemma n_le_s (N n : ℕ) : n ≤ s N n := by |
| have := s_zero_add N n |
| omega |
| |
| lemma ε_pos (b N n : ℕ) (hb : 0 < b) : 0 < ε b N n := by |
| unfold ε |
| positivity |
| |
| lemma ε_nonneg (b N n : ℕ) (hb : 0 < b) : 0 ≤ ε b N n := |
| (ε_pos b N n hb).le |
| |
| lemma ε_strictAnti (b N : ℕ) (hb : 1 < b) : StrictAnti (ε b N) := by |
| intro n m hnm |
| unfold ε |
| apply inv_strictAnti₀ _ _ |
| · positivity |
| · exact pow_lt_pow_right₀ (by exact_mod_cast hb) (strictMono_s N hnm) |
| |
| lemma ε_le_geometric (b N n : ℕ) (hb : 2 ≤ b) : |
| ε b N n ≤ (1 / 2 : ℝ) ^ n := by |
| have hb' : (2 : ℝ) ≤ b := by exact_mod_cast hb |
| unfold ε |
| rw [one_div, inv_pow] |
| apply inv_anti₀ (by positivity) |
| calc |
| (2 : ℝ) ^ n ≤ (2 : ℝ) ^ s N n := pow_le_pow_right₀ (by norm_num) (n_le_s N n) |
| _ ≤ (b : ℝ) ^ s N n := pow_le_pow_left₀ (by norm_num) hb' _ |
|
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| lemma ε_le_initial_geometric (b N n : ℕ) (hb : 2 ≤ b) : |
| ε b N n ≤ ε b N 0 * (1 / 2 : ℝ) ^ n := by |
| have hb' : (2 : ℝ) ≤ b := by exact_mod_cast hb |
| have hp : (b : ℝ) ^ s N 0 * 2 ^ n ≤ (b : ℝ) ^ s N n := by |
| calc |
| (b : ℝ) ^ s N 0 * 2 ^ n ≤ (b : ℝ) ^ s N 0 * (b : ℝ) ^ n := by |
| gcongr |
| _ = (b : ℝ) ^ (s N 0 + n) := (pow_add _ _ _).symm |
| _ ≤ (b : ℝ) ^ s N n := pow_le_pow_right₀ (by linarith) (s_zero_add N n) |
| simpa only [ε, one_div, inv_pow, mul_inv_rev, mul_comm] using |
| (inv_anti₀ (by positivity : (0 : ℝ) < (b : ℝ) ^ s N 0 * 2 ^ n) hp) |
| |
| lemma summable_ε (b N : ℕ) (hb : 2 ≤ b) : Summable (ε b N) := by |
| exact Summable.of_nonneg_of_le |
| (fun n => ε_nonneg b N n (by omega)) |
| (fun n => ε_le_geometric b N n hb) summable_geometric_two |
| |
| lemma total_memory_le (b N : ℕ) (hb : 2 ≤ b) : |
| (∑' n : ℕ, ε b N n) ≤ 2 * ε b N 0 := by |
| have hg := summable_geometric_two.mul_left (ε b N 0) |
| have h := (summable_ε b N hb).tsum_le_tsum |
| (fun n => ε_le_initial_geometric b N n hb) hg |
| calc |
| (∑' n : ℕ, ε b N n) ≤ ∑' n : ℕ, ε b N 0 * (1 / 2 : ℝ) ^ n := h |
| _ = 2 * ε b N 0 := by |
| rw [tsum_mul_left, tsum_geometric_of_norm_lt_one (by norm_num : ‖(1 / 2 : ℝ)‖ < 1)] |
| ring |
|
|
| lemma initial_memory_small (b N : ℕ) (hb : 4 ≤ b) : 2 * ε b N 0 < 1 / 4 := by |
| have hs : 2 ≤ s N 0 := by have := s_lower N 0; nlinarith |
| have hb' : (4 : ℝ) ≤ b := by exact_mod_cast hb |
| have hpow : (16 : ℝ) ≤ (b : ℝ) ^ s N 0 := by |
| calc |
| (16 : ℝ) = 4 ^ 2 := by norm_num |
| _ ≤ (4 : ℝ) ^ s N 0 := pow_le_pow_right₀ (by norm_num) hs |
| _ ≤ (b : ℝ) ^ s N 0 := pow_le_pow_left₀ (by norm_num) hb' _ |
| have hε : ε b N 0 ≤ (16 : ℝ)⁻¹ := inv_anti₀ (by norm_num) hpow |
| norm_num at hε |
| linarith |
|
|
| lemma derivative_cost_bound (b N r n : ℕ) (hb : 2 ≤ b) (hn : r ≤ n) : |
| ε b N (n + 1) * (b : ℝ) ^ ((r + 1) * (s N n + 1 + K N n)) ≤ ε b N n := by |
| have hb' : (1 : ℝ) ≤ b := by exact_mod_cast (show 1 ≤ b by omega) |
| have h := pow_le_pow_right₀ hb' (derivative_gap N r n hn) |
| have hpos : 0 < (b : ℝ) := by exact_mod_cast (show 0 < b by omega) |
| dsimp [ε] |
| rw [← div_eq_inv_mul, ← one_div] |
| apply (div_le_div_iff₀ (pow_pos hpos _) (pow_pos hpos _)).2 |
| simpa [pow_add, mul_comm] using h |
| |
| lemma summable_weighted_ε (b N J : ℕ) (hb : 2 ≤ b) : |
| Summable (fun n : ℕ => (1 + (n : ℝ)) ^ J * ε b N n) := by |
| have h₀ := summable_pow_mul_geometric_of_norm_lt_one J |
| (by norm_num : ‖(1 / 2 : ℝ)‖ < 1) |
| have h₁ := ((summable_nat_add_iff 1).2 h₀).mul_left (2 : ℝ) |
| have h₂ : Summable (fun n : ℕ => (1 + (n : ℝ)) ^ J * (1 / 2 : ℝ) ^ n) := by |
| apply h₁.congr |
| intro n |
| simp only [Nat.cast_add, Nat.cast_one, pow_succ] |
| ring |
| exact Summable.of_nonneg_of_le |
| (fun n => mul_nonneg (by positivity) (ε_nonneg b N n (by omega))) |
| (fun n => mul_le_mul_of_nonneg_left (ε_le_geometric b N n hb) (by positivity)) h₂ |
|
|
| lemma weighted_ε_tendsto_zero (b N J : ℕ) (hb : 2 ≤ b) : |
| Filter.Tendsto (fun n : ℕ => (1 + (n : ℝ)) ^ J * ε b N n) |
| Filter.atTop (𝓝 0) := |
| (summable_weighted_ε b N J hb).tendsto_atTop_zero |
|
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| lemma weighted_ε_bounded (b N J : ℕ) (hb : 2 ≤ b) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, (1 + (n : ℝ)) ^ J * ε b N n ≤ C := by |
| refine ⟨∑' n : ℕ, (1 + (n : ℝ)) ^ J * ε b N n, ?_, ?_⟩ |
| · apply tsum_nonneg |
| intro n |
| exact mul_nonneg (by positivity) (ε_nonneg b N n (by omega)) |
| · intro n |
| apply (summable_weighted_ε b N J hb).le_tsum |
| intro j _ |
| exact mul_nonneg (by positivity) (ε_nonneg b N j (by omega)) |
|
|
| end Scales |
| end AlternatingNS |
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| end OAI |
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