| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Filter | |
| lemma bounded_geometric_absorption_general (B : ℝ) (hB : 1≤B) | |
| (s : ℕ → ℝ) (M A : ℝ) (hA : 0≤A) (hM : ∀ n, s n≤M) | |
| (hr : ∀ n, s n≤(1/(2*B))*s (n+1)+A*B^n) : s 0≤2*A := by | |
| have hBp : 0<B := lt_of_lt_of_le zero_lt_one hB | |
| have hθ : 0≤1/(2*B) := by positivity | |
| have hθ1 : 1/(2*B)<1 := (div_lt_one (by positivity)).mpr (by linarith) | |
| have hprod : (1/(2*B))*B=(1/2:ℝ) := by field_simp | |
| have htel : ∀ n, s 0≤(1/(2*B):ℝ)^n*s n+2*A*(1-(1/2:ℝ)^n) := by | |
| intro n | |
| induction n with | |
| | zero => simp | |
| | succ n ih => | |
| have hz := mul_le_mul_of_nonneg_left (hr n) (pow_nonneg hθ n) | |
| have hp : (1/(2*B):ℝ)^n*B^n=(1/2:ℝ)^n := by rw [←mul_pow,hprod] | |
| rw [pow_succ,pow_succ] | |
| nlinarith | |
| have hh (n : ℕ) : s 0≤(1/(2*B):ℝ)^n*M+2*A := by | |
| have := mul_le_mul_of_nonneg_left (hM n) (pow_nonneg hθ n) | |
| have hz : 0≤2*A*(1/2:ℝ)^n := by positivity | |
| nlinarith [htel n] | |
| have hl := ((tendsto_pow_atTop_nhds_zero_of_lt_one hθ hθ1).mul_const M).add_const (2*A) | |
| simp only [zero_mul,zero_add] at hl | |
| exact ge_of_tendsto hl (Eventually.of_forall hh) | |
| end | |
| end OAI | |