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f61e547 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import Mathlib
namespace OAI
noncomputable section
open Filter
open scoped Topology
namespace Problem367
namespace CriticalBranching
def step (b : ℕ) (p q : ℝ) : ℝ := 1 - (1 - p * q) ^ b
def survival (b : ℕ) (p : ℝ) : ℕ → ℝ
| 0 => 1
| n + 1 => step b p (survival b p n)
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
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)
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
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) _
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
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
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
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
end CriticalBranching
end Problem367
end
end OAI
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