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namespace OAI
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation
lemma sparse_polynomial_doubling_scales (f : ℕ → ℕ) (hf : Monotone f)
(hpos : ∀ n, 1≤f n) (d : ℕ) (hpoly : ∀ R, ∃ n≥R, f n≤n^d) :
∀ R : ℕ, ∃ n≥R, f (5*n)≤5^(d+1)*f n := by
intro R
by_contra! hbad
let A := max R 1
have hA : 0<A := by dsimp [A]; omega
have hRA : R≤A := le_max_left _ _
have hgrow : ∀ k : ℕ, (5^(d+1))^k≤f (5^k*A) := by
intro k
induction k with
| zero => simpa using hpos A
| succ k ih =>
have hbase : R≤5^k*A := hRA.trans (by
have : 1≤5^k := Nat.one_le_pow k 5 (by omega)
nlinarith)
have hh := (hbad (5^k*A) hbase).le
calc
(5^(d+1))^(k+1) = 5^(d+1)*(5^(d+1))^k := by rw [pow_succ]; ring
_ ≤ 5^(d+1)*f (5^k*A) := Nat.mul_le_mul_left _ ih
_ ≤ f (5*(5^k*A)) := hh
_ = f (5^(k+1)*A) := by congr 1; rw [pow_succ]; ring
obtain ⟨n,hn,hnpoly⟩ := hpoly (5*A*((5*A)^d+1))
have hnA : 5*((5*A)^d+1)≤n/A := by
apply (Nat.le_div_iff_mul_le hA).mpr
nlinarith
have hnApos : n/A≠0 := by omega
let k := Nat.log 5 (n/A)
let P := 5^k
have hP : P≤n/A := Nat.pow_log_le_self 5 hnApos
have hP' : n/A<5*P := by
have hh := Nat.lt_pow_of_log_lt (b := 5) (x := k+1) (y := n/A) (by omega)
(show Nat.log 5 (n/A)<k+1 by omega)
simpa only [pow_succ,Nat.mul_comm] using hh
have hPL : (5*A)^d<P := by nlinarith
have hPA : P*A≤n := (Nat.le_div_iff_mul_le hA).mp hP
have hnupper : n<(5*P)*A := (Nat.div_lt_iff_lt_mul hA).mp hP'
have hPP : 5^((d+1)*k)≤f n := by
rw [pow_mul]
exact (hgrow k).trans (hf hPA)
have hform : 5^((d+1)*k)=P^d*P := by
dsimp [P]
rw [←pow_mul,←pow_add]
congr 1
ring
rw [hform] at hPP
have hupper : n^d≤P^d*(5*A)^d := by
have hh := Nat.pow_le_pow_left hnupper.le d
convert hh using 1
ring
have hPpos : 0<P^d := pow_pos (by dsimp [P]; positivity) d
have hlt : P^d*(5*A)^d<P^d*P := Nat.mul_lt_mul_of_pos_left hPL hPpos
exact (not_lt_of_ge (hPP.trans (hnpoly.trans hupper))) hlt
end CriticalPercolation
end
end
end OAI
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