lean/OAI/LinearAlgebra/MatrixMultiplication/Entropy/ComplexSeparationAsymptotics.lean

import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexSeparationCounting
import Mathlib.Data.Nat.Find
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Order.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
import Mathlib.Tactic.GCongr

namespace OAI

/-! Finite entropy, rate estimates and ordered asymptotic limits. -/

noncomputable section

namespace MatrixMultiplication.Foundation.Separation

open Filter

private theorem scale_exists (K : ℕ) : ∃ n : ℕ, K ≤ 2 ^ (n ^ 2) := by
  refine ⟨K + 1, ?_⟩
  have hK : K < 2 ^ K := Nat.lt_pow_self (by decide : 1 < 2)
  exact hK.le.trans (Nat.pow_le_pow_right (by decide) (by nlinarith))

noncomputable def gridScale (K : ℕ) : ℕ := Nat.find (scale_exists K)

theorem population_le_scale (K : ℕ) : K ≤ 2 ^ ((gridScale K) ^ 2) :=
  Nat.find_spec (scale_exists K)

theorem previous_scale_lt_population {K : ℕ} (hn : 0 < gridScale K) :
    2 ^ ((gridScale K - 1) ^ 2) < K := by
  exact lt_of_not_ge (Nat.find_min (scale_exists K) (Nat.sub_lt hn (by decide)))

theorem scale_pos {K : ℕ} (hK : 1 < K) : 0 < gridScale K := by
  have h := population_le_scale K
  by_contra hn
  have hz : gridScale K = 0 := Nat.eq_zero_of_not_pos hn
  simp only [hz, zero_pow (by decide : 2 ≠ 0), pow_zero] at h
  exact (not_le_of_gt hK) h

theorem scale_gt_of_population_gt {K n : ℕ} (hK : 2 ^ (n ^ 2) < K) :
    n < gridScale K := by
  by_contra hn
  have hscale : gridScale K ≤ n := Nat.le_of_not_gt hn
  have hp : 2 ^ ((gridScale K) ^ 2) ≤ 2 ^ (n ^ 2) :=
    Nat.pow_le_pow_right (by decide) (Nat.pow_le_pow_left hscale 2)
  exact (not_le_of_gt hK) ((population_le_scale K).trans hp)

theorem gridScale_monotone : Monotone gridScale := by
  intro K L hKL
  exact Nat.find_min' (scale_exists K) (hKL.trans (population_le_scale L))

def gridDimension (K : ℕ) : ℕ := gridScale K + 4

def gridWidth (K : ℕ) : ℕ := 2 ^ (gridScale K + 2)

def gridGroupOrder (K : ℕ) : ℕ := (4 * gridWidth K) ^ gridDimension K

theorem gridDimension_pos (K : ℕ) : 0 < gridDimension K := by
  simp [gridDimension]

theorem gridWidth_ge_two (K : ℕ) : 2 ≤ gridWidth K := by
  simpa only [gridWidth, pow_one] using
    (Nat.pow_le_pow_right (by decide : 0 < 2) (by omega : 1 ≤ gridScale K + 2))

private theorem dimension_le_power (n : ℕ) : n + 4 ≤ 2 ^ (n + 2) := by
  induction n with
  | zero => norm_num
  | succ n ih =>
    have hp : 2 ^ (n + 2 + 1) = 2 ^ (n + 2) * 2 := pow_succ _ _
    have he : n + 1 + 2 = n + 2 + 1 := by omega
    rw [he, hp]
    omega

theorem grid_population_bound (K : ℕ) :
    (gridDimension K * gridWidth K ^ 2 + 1) * K ≤
      gridWidth K ^ gridDimension K := by
  let n := gridScale K
  have hd : n + 4 ≤ 2 ^ (n + 2) := dimension_le_power n
  have hq : 0 < 2 ^ (n + 2) := by positivity
  have hcolorpos : 0 < (n + 4) * (2 ^ (n + 2)) ^ 2 := by positivity
  have hcolor : (n + 4) * (2 ^ (n + 2)) ^ 2 + 1 ≤ 2 ^ (3 * n + 7) := by
    calc
      (n + 4) * (2 ^ (n + 2)) ^ 2 + 1 ≤
          2 * ((n + 4) * (2 ^ (n + 2)) ^ 2) := by nlinarith
      _ ≤ 2 * (2 ^ (n + 2) * (2 ^ (n + 2)) ^ 2) := by gcongr
      _ = 2 ^ (3 * n + 7) := by
        rw [← pow_mul, ← pow_add, ← pow_succ']
        congr 1
        omega
  calc
    (gridDimension K * gridWidth K ^ 2 + 1) * K ≤
        2 ^ (3 * n + 7) * 2 ^ (n ^ 2) :=
      Nat.mul_le_mul hcolor (population_le_scale K)
    _ = 2 ^ (3 * n + 7 + n ^ 2) := (pow_add _ _ _).symm
    _ ≤ 2 ^ ((n + 2) * (n + 4)) :=
      Nat.pow_le_pow_right (by decide) (by nlinarith)
    _ = gridWidth K ^ gridDimension K := by
      simp only [gridWidth, gridDimension, n, pow_mul]

theorem gridGroupOrder_eq (K : ℕ) :
    gridGroupOrder K = 2 ^ ((gridScale K + 4) ^ 2) := by
  unfold gridGroupOrder gridWidth gridDimension
  have hb : 4 * 2 ^ (gridScale K + 2) = 2 ^ (gridScale K + 4) := by
    calc
      4 * 2 ^ (gridScale K + 2) = 2 ^ 2 * 2 ^ (gridScale K + 2) := by norm_num
      _ = 2 ^ (2 + (gridScale K + 2)) := (pow_add _ _ _).symm
      _ = 2 ^ (gridScale K + 4) := by congr 1; omega
  rw [hb, ← pow_mul, pow_two]

theorem gridGroupOrder_pos (K : ℕ) : 0 < gridGroupOrder K := by
  rw [gridGroupOrder_eq]
  positivity

theorem population_le_gridGroupOrder (K : ℕ) : K ≤ gridGroupOrder K := by
  rw [gridGroupOrder_eq]
  exact (population_le_scale K).trans
    (Nat.pow_le_pow_right (by decide)
      (Nat.pow_le_pow_left (Nat.le_add_right (gridScale K) 4) 2))

theorem gridGroupOrder_monotone : Monotone gridGroupOrder := by
  intro K L hKL
  rw [gridGroupOrder_eq, gridGroupOrder_eq]
  exact Nat.pow_le_pow_right (by decide)
    (Nat.pow_le_pow_left (Nat.add_le_add_right (gridScale_monotone hKL) 4) 2)

theorem asymptotic_grid_tags_slices (K : ℕ) :
    ∃ tags : Finset (Grid (gridDimension K) (gridWidth K)), tags.card = K ∧
      (∃ radius : Fin (gridDimension K * gridWidth K ^ 2 + 1),
        ∀ t ∈ tags, normColor t = radius) ∧
      ∀ t ∈ tags, ∃ slice : Finset (Grid (gridDimension K) (gridWidth K)),
        slice.card = K ∧ ∃ level : Fin (gridDimension K * gridWidth K ^ 2 + 1),
          ∀ u ∈ slice, dotColor t u = level :=
  grid_tags_slices (grid_population_bound K) (grid_population_bound K)

theorem log_gridGroupOrder (K : ℕ) :
    Real.log (gridGroupOrder K : ℝ) = ((gridScale K : ℝ) + 4) ^ 2 * Real.log 2 := by
  rw [gridGroupOrder_eq, Nat.cast_pow, Real.log_pow]
  norm_cast

theorem log_population_le_log_gridGroupOrder {K : ℕ} (hK : 0 < K) :
    Real.log (K : ℝ) ≤ Real.log (gridGroupOrder K : ℝ) := by
  exact Real.log_le_log (by exact_mod_cast hK)
    (by exact_mod_cast population_le_gridGroupOrder K)

theorem previous_scale_log_lt {K : ℕ} (hK : 1 < K) :
    ((gridScale K : ℝ) - 1) ^ 2 * Real.log 2 < Real.log (K : ℝ) := by
  have hn : 1 ≤ gridScale K := scale_pos hK
  have hp : (0 : ℝ) < (2 : ℝ) ^ ((gridScale K - 1) ^ 2) := by positivity
  have hc : (2 : ℝ) ^ ((gridScale K - 1) ^ 2) < (K : ℝ) := by
    exact_mod_cast previous_scale_lt_population (scale_pos hK)
  have h := Real.log_lt_log hp hc
  simpa only [Real.log_pow, Nat.cast_pow, Nat.cast_sub hn, Nat.cast_one] using h

theorem log_gridGroupOrder_le_of_scale {K : ℕ} {ε : ℝ}
    (hK : 1 < K) (hε : 0 ≤ ε)
    (hscale : 10 * (gridScale K : ℝ) + 15 ≤
      ε * ((gridScale K : ℝ) - 1) ^ 2) :
    Real.log (gridGroupOrder K : ℝ) ≤ (1 + ε) * Real.log (K : ℝ) := by
  have hlog2 : 0 < Real.log 2 := Real.log_pos (by norm_num)
  have hpoly : ((gridScale K : ℝ) + 4) ^ 2 ≤
      (1 + ε) * ((gridScale K : ℝ) - 1) ^ 2 := by nlinarith
  calc
    Real.log (gridGroupOrder K : ℝ) =
        ((gridScale K : ℝ) + 4) ^ 2 * Real.log 2 := log_gridGroupOrder K
    _ ≤ ((1 + ε) * ((gridScale K : ℝ) - 1) ^ 2) * Real.log 2 :=
      mul_le_mul_of_nonneg_right hpoly hlog2.le
    _ = (1 + ε) * (((gridScale K : ℝ) - 1) ^ 2 * Real.log 2) := by ring
    _ ≤ (1 + ε) * Real.log (K : ℝ) :=
      mul_le_mul_of_nonneg_left (previous_scale_log_lt hK).le (by positivity)

private theorem scale_error_bound {n ε : ℝ} (hn : 2 ≤ n) (hε : 0 ≤ ε)
    (hlarge : 72 ≤ ε * n) : 10 * n + 15 ≤ ε * (n - 1) ^ 2 := by
  have hsq : n ^ 2 ≤ 4 * (n - 1) ^ 2 := by nlinarith
  have hmul := mul_le_mul_of_nonneg_left hsq hε
  have hlarge' := mul_le_mul_of_nonneg_right hlarge (by linarith : 0 ≤ n)
  nlinarith

theorem eventually_log_gridGroupOrder_le {ε : ℝ} (hε : 0 < ε) :
    ∀ᶠ K : ℕ in atTop,
      Real.log (gridGroupOrder K : ℝ) ≤ (1 + ε) * Real.log (K : ℝ) := by
  obtain ⟨m : ℕ, hm⟩ := exists_nat_gt (max 2 (72 / ε))
  refine eventually_atTop.2 ⟨2 ^ (m ^ 2) + 1, ?_⟩
  intro K hK
  have hKm : 2 ^ (m ^ 2) < K := by omega
  have hn := scale_gt_of_population_gt hKm
  have hnR : (m : ℝ) < (gridScale K : ℝ) := by exact_mod_cast hn
  have hm2 : 2 < (m : ℝ) := (le_max_left _ _).trans_lt hm
  have hmε : 72 / ε < (m : ℝ) := (le_max_right _ _).trans_lt hm
  have hnε : 72 ≤ ε * (gridScale K : ℝ) := by
    have hdiv : 72 / ε < (gridScale K : ℝ) := hmε.trans hnR
    have hmul := (div_lt_iff₀ hε).1 hdiv
    nlinarith
  have hKone : 1 < K := by
    have hp : 1 ≤ 2 ^ (m ^ 2) := Nat.one_le_pow _ _ (by decide)
    omega
  exact log_gridGroupOrder_le_of_scale hKone hε.le
    (scale_error_bound (by linarith) hε.le hnε)

theorem uniform_log_gridGroupOrder_le {ε : ℝ} (hε : 0 < ε) :
    ∃ C : ℝ, 0 ≤ C ∧ ∀ K : ℕ,
      Real.log (gridGroupOrder K : ℝ) ≤ (1 + ε) * Real.log (K : ℝ) + C := by
  obtain ⟨K₀, hK₀⟩ := eventually_atTop.1 (eventually_log_gridGroupOrder_le hε)
  refine ⟨Real.log (gridGroupOrder K₀ : ℝ), Real.log_natCast_nonneg _, ?_⟩
  intro K
  by_cases hK : K₀ ≤ K
  · exact (hK₀ K hK).trans (le_add_of_nonneg_right (Real.log_natCast_nonneg _))
  · have horder : gridGroupOrder K ≤ gridGroupOrder K₀ :=
      gridGroupOrder_monotone (Nat.le_of_lt (Nat.lt_of_not_ge hK))
    have hpos : (0 : ℝ) < (gridGroupOrder K : ℝ) := by
      rw [gridGroupOrder_eq, Nat.cast_pow]
      positivity
    have hcast : (gridGroupOrder K : ℝ) ≤ (gridGroupOrder K₀ : ℝ) := by
      exact_mod_cast horder
    have hlog := Real.log_le_log hpos hcast
    have hnonneg : 0 ≤ (1 + ε) * Real.log (K : ℝ) :=
      mul_nonneg (by linarith) (Real.log_natCast_nonneg K)
    linarith

theorem tendsto_log_gridGroupOrder_div_log_population :
    Tendsto (fun K : ℕ => Real.log (gridGroupOrder K : ℝ) / Real.log (K : ℝ))
      atTop (nhds 1) := by
  refine tendsto_order.2 ⟨?_, ?_⟩
  · intro a ha
    refine eventually_atTop.2 ⟨2, ?_⟩
    intro K hK
    have hKone : 1 < K := by omega
    have hlog : 0 < Real.log (K : ℝ) :=
      Real.log_pos (by exact_mod_cast hKone)
    have hle := log_population_le_log_gridGroupOrder (by omega : 0 < K)
    exact ha.trans_le ((le_div_iff₀ hlog).2 (by simpa using hle))
  · intro b hb
    have he : 0 < (b - 1) / 2 := by linarith
    filter_upwards [eventually_log_gridGroupOrder_le he,
      (eventually_atTop.2 ⟨2, fun K (hK : 2 ≤ K) => hK⟩)] with K hcost hK
    have hlog : 0 < Real.log (K : ℝ) :=
      Real.log_pos (by exact_mod_cast (show 1 < K by omega))
    have hratio : Real.log (gridGroupOrder K : ℝ) / Real.log (K : ℝ) ≤
        1 + (b - 1) / 2 := (div_le_iff₀ hlog).2 hcost
    linarith

theorem tendsto_log_gridGroupOrder_div_nat {K : ℕ → ℕ} {H : ℝ}
    (hK : ∀ N, 0 < K N) (hH : 0 ≤ H)
    (hpop : Tendsto (fun N => Real.log (K N : ℝ) / (N : ℝ)) atTop (nhds H)) :
    Tendsto (fun N => Real.log (gridGroupOrder (K N) : ℝ) / (N : ℝ))
      atTop (nhds H) := by
  refine tendsto_order.2 ⟨?_, ?_⟩
  · intro a ha
    filter_upwards [(tendsto_order.1 hpop).1 a ha,
      (eventually_atTop.2 ⟨1, fun N (hN : 1 ≤ N) => hN⟩)] with N haN hN
    have hNpos : (0 : ℝ) < (N : ℝ) := by exact_mod_cast hN
    exact haN.trans_le (div_le_div_of_nonneg_right
      (log_population_le_log_gridGroupOrder (hK N)) hNpos.le)
  · intro b hb
    let ε := (b - H) / (2 * (H + 1))
    have hdenom : 0 < 2 * (H + 1) := by linarith
    have hε : 0 < ε := div_pos (by linarith) hdenom
    obtain ⟨C, hC, hbound⟩ := uniform_log_gridGroupOrder_le hε
    have hεeq : ε * (2 * (H + 1)) = b - H := div_mul_cancel₀ _ hdenom.ne'
    have hlimitlt : (1 + ε) * H < b := by nlinarith
    have hconstant : Tendsto (fun N : ℕ => C / (N : ℝ)) atTop (nhds 0) :=
      tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop
    have hupper : Tendsto
        (fun N : ℕ => (1 + ε) * (Real.log (K N : ℝ) / (N : ℝ)) + C / (N : ℝ))
        atTop (nhds ((1 + ε) * H)) := by
      simpa using (hpop.const_mul (1 + ε)).add hconstant
    filter_upwards [(tendsto_order.1 hupper).2 b hlimitlt,
      (eventually_atTop.2 ⟨1, fun N (hN : 1 ≤ N) => hN⟩)] with N hbN hN
    have hNpos : (0 : ℝ) < (N : ℝ) := by exact_mod_cast hN
    have hle := div_le_div_of_nonneg_right (hbound (K N)) hNpos.le
    have heq : ((1 + ε) * Real.log (K N : ℝ) + C) / (N : ℝ) =
        (1 + ε) * (Real.log (K N : ℝ) / (N : ℝ)) + C / (N : ℝ) := by ring
    rw [heq] at hle
    exact hle.trans_lt hbN

theorem log_nat_prod {I : Type*} (s : Finset I) (a : I → ℕ)
    (ha : ∀ i ∈ s, 0 < a i) :
    Real.log ((∏ i ∈ s, a i : ℕ) : ℝ) = ∑ i ∈ s, Real.log (a i : ℝ) := by
  rw [Nat.cast_prod]
  exact Real.log_prod (s := s) (f := fun i => (a i : ℝ)) fun i hi =>
    Nat.cast_ne_zero.mpr (Nat.ne_of_gt (ha i hi))

theorem tendsto_log_nat_prod_div_nat {I : Type*} (s : Finset I)
    {K : I → ℕ → ℕ} {H : I → ℝ}
    (hK : ∀ i ∈ s, ∀ N, 0 < K i N)
    (hpop : ∀ i ∈ s,
      Tendsto (fun N => Real.log (K i N : ℝ) / (N : ℝ)) atTop (nhds (H i))) :
    Tendsto (fun N => Real.log ((∏ i ∈ s, K i N : ℕ) : ℝ) / (N : ℝ))
      atTop (nhds (∑ i ∈ s, H i)) := by
  have hsum := tendsto_finsetSum s hpop
  apply hsum.congr
  intro N
  rw [log_nat_prod s (fun i => K i N) (fun i hi => hK i hi N)]
  simp only [div_eq_mul_inv, Finset.sum_mul]

theorem tendsto_log_gridGroupOrder_prod_div_nat {I : Type*} (s : Finset I)
    {K : I → ℕ → ℕ} {H : I → ℝ}
    (hK : ∀ i ∈ s, ∀ N, 0 < K i N) (hH : ∀ i ∈ s, 0 ≤ H i)
    (hpop : ∀ i ∈ s,
      Tendsto (fun N => Real.log (K i N : ℝ) / (N : ℝ)) atTop (nhds (H i))) :
    Tendsto (fun N => Real.log ((∏ i ∈ s, gridGroupOrder (K i N) : ℕ) : ℝ) / (N : ℝ))
      atTop (nhds (∑ i ∈ s, H i)) := by
  exact tendsto_log_nat_prod_div_nat s (fun i hi N => gridGroupOrder_pos (K i N))
    (fun i hi => tendsto_log_gridGroupOrder_div_nat (hK i hi) (hH i hi) (hpop i hi))

theorem tendsto_log_nat_weighted_prod_div_nat {I : Type*} (s : Finset I) (w : I → ℕ)
    {K : I → ℕ → ℕ} {H : I → ℝ}
    (hK : ∀ i ∈ s, ∀ N, 0 < K i N)
    (hpop : ∀ i ∈ s,
      Tendsto (fun N => Real.log (K i N : ℝ) / (N : ℝ)) atTop (nhds (H i))) :
    Tendsto (fun N => Real.log ((∏ i ∈ s, (K i N) ^ w i : ℕ) : ℝ) / (N : ℝ))
      atTop (nhds (∑ i ∈ s, (w i : ℝ) * H i)) := by
  apply tendsto_log_nat_prod_div_nat s
  · intro i hi N
    exact pow_pos (hK i hi N) _
  · intro i hi
    simpa only [Nat.cast_pow, Real.log_pow, mul_div_assoc] using
      (hpop i hi).const_mul (w i : ℝ)

theorem tendsto_log_gridGroupOrder_weighted_prod_div_nat {I : Type*}
    (s : Finset I) (w : I → ℕ) {K : I → ℕ → ℕ} {H : I → ℝ}
    (hK : ∀ i ∈ s, ∀ N, 0 < K i N) (hH : ∀ i ∈ s, 0 ≤ H i)
    (hpop : ∀ i ∈ s,
      Tendsto (fun N => Real.log (K i N : ℝ) / (N : ℝ)) atTop (nhds (H i))) :
    Tendsto
      (fun N => Real.log ((∏ i ∈ s, gridGroupOrder (K i N) ^ w i : ℕ) : ℝ) / (N : ℝ))
      atTop (nhds (∑ i ∈ s, (w i : ℝ) * H i)) := by
  exact tendsto_log_nat_weighted_prod_div_nat s w
    (fun i hi N => gridGroupOrder_pos (K i N))
    (fun i hi => tendsto_log_gridGroupOrder_div_nat (hK i hi) (hH i hi) (hpop i hi))

end MatrixMultiplication.Foundation.Separation

end

end OAI