import Mathlib

namespace OAI

section
noncomputable section
open scoped Classical

namespace CriticalPercolation.HarmonicAnalysis

lemma sparse_real_doubling (f : ℕ → ℝ) (hf : Monotone f)
    (R₀ : ℕ) (hpos : ∀ n≥R₀, 0<f n) (C : ℝ) (hC : 0≤C)
    (d b : ℕ) (hb : 2≤b) (hpoly : ∀ R, ∃ n≥R, f n≤C*(n:ℝ)^d) :
    ∀ R : ℕ, ∃ n≥max R R₀, f (b*n)≤(b:ℝ)^(d+1)*f n := by
  intro R
  by_contra! hbad
  let a := max (max R R₀) 1
  have ha : 0<a := by dsimp [a]; omega
  have haR : max R R₀≤a := le_max_left _ _
  have hfapos : 0<f a := hpos a ((le_max_right R R₀).trans haR)
  have hgrow : ∀ k : ℕ, ((b:ℝ)^(d+1))^k*f a≤f (b^k*a) := by
    intro k
    induction k with
    | zero => simp
    | succ k ih =>
      have hr : max R R₀≤b^k*a := haR.trans (by
        have hh : 1≤b^k := Nat.one_le_pow k b (by omega)
        nlinarith)
      have hh := (hbad (b^k*a) hr).le
      calc
        _ = (b:ℝ)^(d+1)*(((b:ℝ)^(d+1))^k*f a) := by rw [pow_succ]; ring
        _ ≤ (b:ℝ)^(d+1)*f (b^k*a) := mul_le_mul_of_nonneg_left ih (by positivity)
        _ ≤ f (b*(b^k*a)) := hh
        _ = _ := by congr 1; rw [pow_succ]; ring
  obtain ⟨N,hN⟩ := exists_nat_gt (C*((b*a:ℕ):ℝ)^d/f a)
  have hNF : C*((b*a:ℕ):ℝ)^d<(N:ℝ)*f a := (div_lt_iff₀ hfapos).mp hN
  obtain ⟨n,hn,hnpoly⟩ := hpoly (b*a*(N+1))
  have hnA : b*(N+1)≤n/a := (Nat.le_div_iff_mul_le ha).mpr (by nlinarith)
  have hnAp : n/a≠0 := by nlinarith
  let k := Nat.log b (n/a)
  let P := b^k
  have hP : P≤n/a := Nat.pow_log_le_self b hnAp
  have hP' : n/a<b*P := by
    have hh := Nat.lt_pow_of_log_lt (b := b) (x := k+1) (y := n/a) hb
      (show Nat.log b (n/a)<k+1 by omega)
    simpa only [pow_succ,Nat.mul_comm] using hh
  have hNP : N<P := by nlinarith
  have hNP' : (N:ℝ)<P := by exact_mod_cast hNP
  have hPA : P*a≤n := (Nat.le_div_iff_mul_le ha).mp hP
  have hnupper : n<(b*P)*a := (Nat.div_lt_iff_lt_mul ha).mp hP'
  have hlow : (P:ℝ)^d*((P:ℝ)*f a)≤f n := by
    have hh := (hgrow k).trans (hf hPA)
    have he : (P:ℝ)^d*((P:ℝ)*f a) = ((b:ℝ)^(d+1))^k*f a := by
      dsimp [P]
      push_cast
      rw [←mul_assoc,←pow_succ,←pow_mul,←pow_mul,Nat.mul_comm k (d+1)]
    rw [he]
    exact hh
  have hupper : f n≤(P:ℝ)^d*(C*((b*a:ℕ):ℝ)^d) := by
    have hnupper' : (n:ℝ)≤((b*P)*a:ℕ) := by exact_mod_cast hnupper.le
    have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (Nat.cast_nonneg _) hnupper' d) hC
    apply hnpoly.trans
    convert hh using 1
    push_cast
    ring
  have hPpos : (0:ℝ)<(P:ℝ)^d := by
    apply pow_pos
    exact_mod_cast (show 0<P by dsimp [P]; positivity)
  have hsmall : C*((b*a:ℕ):ℝ)^d<(P:ℝ)*f a :=
    hNF.trans (mul_lt_mul_of_pos_right hNP' hfapos)
  exact (not_lt_of_ge (hlow.trans hupper)) (mul_lt_mul_of_pos_left hsmall hPpos)

lemma exists_local_ratio (f : ℕ → ℝ) (L : ℕ) (hL : 0<L)
    (C : ℝ) (hC : 0<C) (hbound : f L≤C^L*f 0) :
    ∃ i<L, f (i+1)≤C*f i := by
  by_contra! hbad
  have hgrow : ∀ i≤L, C^i*f 0≤f i := by
    intro i hi
    induction i with
    | zero => simp
    | succ i ih =>
      have hh := ih (by omega)
      calc
        _ = C*(C^i*f 0) := by rw [pow_succ]; ring
        _ ≤ C*f i := mul_le_mul_of_nonneg_left hh hC.le
        _ ≤ _ := (hbad i (by omega)).le
  have hlast := hbad (L-1) (by omega)
  rw [Nat.sub_add_cancel (by omega : 1≤L)] at hlast
  have hg := mul_le_mul_of_nonneg_left (hgrow (L-1) (by omega)) hC.le
  have he : C*(C^(L-1)*f 0)=C^L*f 0 := by
    rw [←mul_assoc,←pow_succ',Nat.sub_add_cancel (by omega : 1≤L)]
  rw [he] at hg
  exact (not_lt_of_ge hbound) (hg.trans_lt hlast)

end CriticalPercolation.HarmonicAnalysis
end
end

end OAI