| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set |
| open scoped Classical |
|
|
| namespace CriticalPercolation |
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| 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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|
