| import Mathlib |
| import OAI.Probability.SKGap.Model |
|
|
| namespace OAI |
|
|
| namespace SKGap.DiscreteRecurrence |
| open Real |
|
|
| lemma affine_upper (u : ℕ→ℝ) {b r : ℝ} (hb : 0≤b) (hr : 0≤r) |
| (hzero : u 0≤0) (K : ℕ) |
| (hstep : ∀k<K,u (k+1)≤(1+b)*u k+r) : |
| u K≤(K:ℝ)*r*exp ((K:ℝ)*b) := by |
| have hh : ∀ k≤K,u k≤(k:ℝ)*r*(1+b)^k := by |
| intro k hk |
| induction k with |
| | zero => simpa using hzero |
| | succ k ih => |
| have hi:=mul_le_mul_of_nonneg_left (ih (Nat.le_of_succ_le hk)) (by linarith : 0≤1+b) |
| have hp : 1≤(1+b)^(k+1) := one_le_pow₀ (by linarith) |
| have hrp:=mul_le_mul_of_nonneg_left hp hr |
| have hs:=hstep k (Nat.lt_of_succ_le hk) |
| simp only [Nat.cast_add,Nat.cast_one,pow_succ] at hrp ⊢ |
| nlinarith |
| have hp:=pow_le_pow_left₀ (by linarith : 0≤1+b) (by simpa only [add_comm] using Real.add_one_le_exp b) K |
| have hm:=mul_le_mul_of_nonneg_left hp (show 0≤(K:ℝ)*r by positivity) |
| rw [←Real.exp_nat_mul] at hm |
| simpa only [mul_comm b] using (hh K le_rfl).trans hm |
|
|
| lemma affine_lower (u : ℕ→ℝ) {r b : ℝ} (hr : 0≤r) (hr1 : r≤1) (hb : 0≤b) |
| (K : ℕ) (hstep : ∀k<K,r*u k-b≤u (k+1)) : |
| r^K*u 0-(K:ℝ)*b≤u K := by |
| induction K with |
| | zero => simp |
| | succ K ih => |
| have hi:=mul_le_mul_of_nonneg_left (ih (fun k hk => hstep k (hk.trans (Nat.lt_succ_self K)))) hr |
| have hs:=hstep K (Nat.lt_succ_self K) |
| have hb':=mul_le_mul_of_nonneg_right hr1 (show 0≤(K:ℝ)*b by positivity) |
| rw [Nat.cast_add,Nat.cast_one,pow_succ] |
| nlinarith |
|
|
| lemma exp_neg_two_le {x : ℝ} (hx : 0≤x) (hx1 : x≤1/2) : |
| exp (-2*x)≤1-x := by |
| have hpos : 0<1-x := by linarith |
| have hi : (1-x)⁻¹≤1+2*x := by |
| rw [inv_eq_one_div,div_le_iff₀ hpos] |
| nlinarith |
| have hl:=Real.one_sub_inv_le_log_of_pos hpos |
| have hh : -2*x≤log (1-x) := by linarith |
| simpa only [Real.exp_log hpos] using Real.exp_le_exp.mpr hh |
|
|
| lemma pow_one_sub_lower {x : ℝ} (hx : 0≤x) (hx1 : x≤1/2) (K : ℕ) : |
| exp (-2*(K:ℝ)*x)≤(1-x)^K := by |
| have hh:=pow_le_pow_left₀ (exp_pos (-2*x)).le (exp_neg_two_le hx hx1) K |
| rw [←Real.exp_nat_mul] at hh |
| have he : (K:ℝ)*(-2*x)= -2*(K:ℝ)*x := by ring |
| rwa [he] at hh |
|
|
| end SKGap.DiscreteRecurrence |
|
|
| end OAI |
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|
