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namespace OAI
/- Prime supply and bounded parameters -/
namespace PowerSaving.Selection
open Finset Real
noncomputable section
def stackN (n : ℕ) := Nat.log 2 (2*n)+1
lemma stack_pos (n : ℕ) : 0< stackN n := Nat.succ_pos _
lemma stack_above (n : ℕ) :
2*n < 2^stackN n :=
Nat.lt_pow_succ_log_self Nat.one_lt_two _
lemma stack_pow (n : ℕ) (hn : 0<n) :
2^stackN n ≤ 4*n := by
have hh : 2^(Nat.log 2 (2*n)) ≤ 2*n := Nat.pow_log_le_self _ (by omega)
rw [stackN,pow_succ]; omega
lemma stack_le (n : ℕ) (hn : 0<n) :
stackN n ≤ 4*n := Nat.lt_two_pow_self.le.trans (stack_pow n hn)
def silo (k : ℕ) := (16*(k+4))^2
lemma silo_big (k : ℕ) : 2 < silo k := by
have hh : 2 < 16*(k+4) := by omega
unfold silo; nlinarith [hh]
lemma harvest (k : ℕ) :
k+2 ≤ Nat.primeCounting (silo k) := by
let q := silo k
let t : ℝ := k+4
let s : ℝ := 16*t
have hq : (q:ℝ)=s^2 := by dsimp [q,silo,s,t]; push_cast; ring
have ht : 4≤t := by simp [t]
have hs : 64≤ s := by unfold s; linarith
have he : 0< s := by linarith
have hh : 1 < (q:ℝ) := by rw [hq]; nlinarith
have HL : log (q:ℝ) ≤ 2*s := by
rw [hq, Real.log_pow]
norm_num
have := Real.log_le_sub_one_of_pos he
linarith
have hk : log (2:ℝ) ≥ 2/3 := by
have hh := Real.le_log_one_add_of_nonneg (show (0:ℝ)≤1 by norm_num)
norm_num at hh ⊢
exact hh
have hu : log ((q:ℝ)+1) ≤ 1+2*s := by
have h1 : (q:ℝ)+1 ≤ 2*q := by linarith
have hy := Real.log_le_log (by linarith : 0<(q:ℝ)+1) h1
rw [Real.log_mul (by norm_num) (by linarith)] at hy
have h2 := Real.log_le_sub_one_of_pos (show (0:ℝ)<2 by norm_num)
linarith
have h3 := Chebyshev.pi_ge q
have HG : (k:ℝ)+2 ≤ Nat.primeCounting q := by
refine le_trans ?_ h3
rw [le_div_iff₀ (Real.log_pos hh)]
calc
_ ≤ t * (2*s) := by
have hg : (k:ℝ)+2 ≤ t := by unfold t; linarith
gcongr
_ ≤ s^2*(2/3)-(1+2*s) := by
have g : s=16*t := rfl
nlinarith
_ ≤ _ := by rw [hq] at hu ⊢; gcongr
exact_mod_cast HG
-- odd primes up to and excluding limit
def Bar (i : ℕ) := Nat.Prime i ∧ i>2
instance (i) : Decidable (Bar i) := inferInstanceAs (Decidable (_ ∧ _))
def bars (p : ℕ) : ℕ := ∑ i ∈ range p, if Bar i then 1 else 0
lemma bars_step (i : ℕ) : bars (i+1)=bars i+(if Bar i then 1 else 0) := sum_range_succ ..
lemma bars_zero : bars 0=0 := by simp [bars]
lemma bars_count (p : ℕ) (hp : 2 ≤ p) :
bars (p+1)+1 = Nat.primeCounting p := by
let Q := range (p+1)
have h (i : ℕ) : (if Bar i then 1 else 0)+(if i=2 then 1 else 0)=
(if i.Prime then 1 else 0) := by
by_cases g : i.Prime
· have ht := g.two_le; simp only [Bar,g,true_and,ite_true]; split_ifs <;> omega
· have hh : i≠2 := by intro H; rw [H] at g; exact g Nat.prime_two
simp [g,hh,Bar]
have hh1 : ∑ i ∈ Q, (if i=2 then 1 else 0)=1 := by
have hg : 2<p+1 := by omega
simp [Q,hg]
conv_lhs => rhs; rw [← hh1]
rw [bars]
dsimp only [Q]
rw [← sum_add_distrib]
simp_rw [h]
rw [← Nat.primesLE_card_eq_primeCounting]
-- filter
rw [← sum_filter]
simp [Nat.primesLE,Nat.primesBelow]
lemma bar_bounds (k : ℕ) :
k < bars (silo k+1) := by
have H := bars_count _ (silo_big k).le
have G := harvest k
omega
end
end PowerSaving.Selection
end OAI
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