| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open Filter |
| open scoped Topology |
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| namespace Problem367 |
| namespace CriticalBranching |
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| def step (b : ℕ) (p q : ℝ) : ℝ := 1 - (1 - p * q) ^ b |
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| def survival (b : ℕ) (p : ℝ) : ℕ → ℝ |
| | 0 => 1 |
| | n + 1 => step b p (survival b p n) |
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| theorem one_sub_pow_quadratic (n : ℕ) {y : ℝ} (_hy0 : 0 ≤ y) (hy1 : y ≤ 1) : |
| 1 - (1 - y) ^ (n + 1) ≤ (n + 1 : ℕ) * y - (n : ℝ) * y ^ 2 := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| have hnon : 0 ≤ 1 - y := sub_nonneg.mpr hy1 |
| calc |
| 1 - (1 - y) ^ (n + 1 + 1) = |
| y + (1 - y) * (1 - (1 - y) ^ (n + 1)) := by rw [pow_succ]; ring |
| _ ≤ y + (1 - y) * ((n + 1 : ℕ) * y - (n : ℝ) * y ^ 2) := by |
| gcongr |
| _ ≤ (n + 1 + 1 : ℕ) * y - (n + 1 : ℕ) * y ^ 2 := by |
| push_cast |
| have h := mul_nonneg (Nat.cast_nonneg n : (0 : ℝ) ≤ n) |
| (mul_nonneg (sq_nonneg y) hnon) |
| nlinarith |
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| theorem step_mem_Icc (b : ℕ) {p q : ℝ} (hp : p ∈ Set.Icc (0 : ℝ) 1) |
| (hq : q ∈ Set.Icc (0 : ℝ) 1) : step b p q ∈ Set.Icc (0 : ℝ) 1 := by |
| have hy1 : p * q ≤ 1 := |
| (mul_le_mul_of_nonneg_right hp.2 hq.1).trans (by simpa using hq.2) |
| have hz0 : 0 ≤ 1 - p * q := sub_nonneg.mpr hy1 |
| have hz1 : 1 - p * q ≤ 1 := by nlinarith [mul_nonneg hp.1 hq.1] |
| constructor |
| · exact sub_nonneg.mpr (pow_le_one₀ hz0 hz1) |
| · exact sub_le_self _ (pow_nonneg hz0 b) |
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| theorem survival_mem_Icc (b : ℕ) {p : ℝ} (hp : p ∈ Set.Icc (0 : ℝ) 1) |
| (n : ℕ) : survival b p n ∈ Set.Icc (0 : ℝ) 1 := by |
| induction n with |
| | zero => simp [survival] |
| | succ n ih => exact step_mem_Icc b hp ih |
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| theorem step_le_quadratic {b : ℕ} (hb : 1 ≤ b) {p q : ℝ} |
| (hp : p ∈ Set.Icc (0 : ℝ) 1) (hq : q ∈ Set.Icc (0 : ℝ) 1) |
| (hcritical : (b : ℝ) * p ≤ 1) : |
| step b p q ≤ q - ((b : ℝ) - 1) * p ^ 2 * q ^ 2 := by |
| obtain ⟨n, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : b ≠ 0) |
| have hy0 : 0 ≤ p * q := mul_nonneg hp.1 hq.1 |
| have hy1 : p * q ≤ 1 := |
| (mul_le_mul_of_nonneg_right hp.2 hq.1).trans (by simpa using hq.2) |
| have h := one_sub_pow_quadratic n hy0 hy1 |
| have hmul := mul_le_mul_of_nonneg_right hcritical hq.1 |
| unfold step |
| simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one] at hmul ⊢ |
| calc |
| 1 - (1 - p * q) ^ (n + 1) ≤ (n + 1 : ℕ) * (p * q) - (n : ℝ) * (p * q) ^ 2 := h |
| _ = ((n : ℝ) + 1) * p * q - ((n : ℝ) + 1 - 1) * p ^ 2 * q ^ 2 := by push_cast; ring |
| _ ≤ q - ((n : ℝ) + 1 - 1) * p ^ 2 * q ^ 2 := sub_le_sub_right (by simpa using hmul) _ |
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| theorem le_reciprocal_of_quadratic_drift {q : ℕ → ℝ} {c : ℝ} |
| (hc : 0 ≤ c) (hq : ∀ n, 0 ≤ q n) (hzero : q 0 ≤ 1) |
| (hstep : ∀ n, q (n + 1) ≤ q n - c * (q n) ^ 2) : |
| ∀ n, q n ≤ 1 / (1 + c * n) := by |
| intro n |
| induction n with |
| | zero => simpa using hzero |
| | succ n ih => |
| have hqn := hq n |
| have hd : 0 < 1 + c * q n := by positivity |
| have hn : 0 < 1 + c * n := by positivity |
| have hn' : 0 < 1 + c * (n + 1 : ℕ) := by positivity |
| have hi := (le_div_iff₀ hn).mp ih |
| calc |
| q (n + 1) ≤ q n / (1 + c * q n) := by |
| apply (le_div_iff₀ hd).mpr |
| have hs := mul_le_mul_of_nonneg_right (hstep n) hd.le |
| have hnon : 0 ≤ c ^ 2 * (q n) ^ 3 := by positivity |
| nlinarith |
| _ ≤ 1 / (1 + c * (n + 1 : ℕ)) := by |
| apply (div_le_div_iff₀ hd hn').mpr |
| push_cast |
| nlinarith |
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| theorem tendsto_zero_of_quadratic_drift {q : ℕ → ℝ} {c : ℝ} |
| (hc : 0 < c) (hq : ∀ n, 0 ≤ q n) (hzero : q 0 ≤ 1) |
| (hstep : ∀ n, q (n + 1) ≤ q n - c * (q n) ^ 2) : |
| Tendsto q atTop (𝓝 0) := by |
| have hb := le_reciprocal_of_quadratic_drift hc.le hq hzero hstep |
| have hinf : Tendsto (fun n : ℕ => 1 + c * (n : ℝ)) atTop atTop := |
| tendsto_const_nhds.add_atTop (tendsto_natCast_atTop_atTop.const_mul_atTop hc) |
| have hz : Tendsto (fun n : ℕ => 1 / (1 + c * n)) atTop (𝓝 (0 : ℝ)) := by |
| simpa only [one_div, Function.comp_def] using tendsto_inv_atTop_zero.comp hinf |
| exact squeeze_zero hq hb hz |
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| theorem survival_le_reciprocal {b : ℕ} (hb : 1 ≤ b) {p : ℝ} |
| (hp : p ∈ Set.Icc (0 : ℝ) 1) (hcritical : (b : ℝ) * p ≤ 1) (n : ℕ) : |
| survival b p n ≤ 1 / (1 + (((b : ℝ) - 1) * p ^ 2) * n) := by |
| apply le_reciprocal_of_quadratic_drift |
| · have hb' : (1 : ℝ) ≤ b := by exact_mod_cast hb |
| positivity |
| · exact fun k => (survival_mem_Icc b hp k).1 |
| · simp [survival] |
| · intro k |
| exact step_le_quadratic hb hp (survival_mem_Icc b hp k) hcritical |
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| theorem survival_tendsto_zero {b : ℕ} (hb : 2 ≤ b) {p : ℝ} |
| (hp : p ∈ Set.Icc (0 : ℝ) 1) (hcritical : (b : ℝ) * p ≤ 1) : |
| Tendsto (survival b p) atTop (𝓝 0) := by |
| by_cases hpzero : p = 0 |
| · subst p |
| apply tendsto_congr' (Filter.eventually_atTop.mpr ⟨1, ?_⟩) |>.mpr tendsto_const_nhds |
| intro n hn |
| obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : n ≠ 0) |
| simp [survival, step] |
| · have hp0 : 0 < p := lt_of_le_of_ne hp.1 (Ne.symm hpzero) |
| apply tendsto_zero_of_quadratic_drift (c := ((b : ℝ) - 1) * p ^ 2) |
| · have hb' : (1 : ℝ) < b := by exact_mod_cast hb |
| positivity |
| · exact fun n => (survival_mem_Icc b hp n).1 |
| · simp [survival] |
| · intro n |
| exact step_le_quadratic (by omega) hp (survival_mem_Icc b hp n) hcritical |
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| end CriticalBranching |
| end Problem367 |
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| end |
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| end OAI |
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