lean/OAI/LinearAlgebra/MatrixFields/Extraction/AuxiliarySeparation.lean

import Mathlib
import OAI.LinearAlgebra.MatrixFields.Entropy.ZeroLeafRates
import OAI.LinearAlgebra.MatrixFields.Tensors.ComplexSeparatedTensorDecomposition

namespace OAI

namespace MatrixAllFields

open scoped BigOperators Topology Polynomial

namespace MatrixMultiplication.Foundation

instance auxiliarySeparation_modulus_neZero (K : ℕ) :
    NeZero (4 * Separation.gridWidth K) :=
  ⟨Nat.mul_ne_zero (by decide)
    (Nat.ne_of_gt (lt_of_lt_of_le (by decide : 0 < 2)
      (Separation.gridWidth_ge_two K)))⟩

section AuxiliarySeparation

variable {X Y Z S : Type*} [Fintype S] [DecidableEq S]

theorem auxiliary_separation_mem_restrictionOrbit
    [Fintype X] [Fintype Y] [Fintype Z]
    (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
    (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
    separatedTensor (U := Fin (Fintype.card S)) lx T ∈ Tensor.restrictionOrbit
      (Tensor.product T (groupTensor (Separation.AuxiliaryGroup
        (Separation.gridDimension (Fintype.card S))
        (Separation.gridWidth (Fintype.card S))))) := by
  exact finite_grid_separation_mem_restrictionOrbit T lx ly
    (Separation.grid_population_bound (Fintype.card S))
    (Separation.grid_population_bound (Fintype.card S)) hlabels

theorem auxiliary_separation_rankAtMost
    (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
    (hT : Tensor.RankAtMost T r)
    (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
    Tensor.RankAtMost (separatedTensor (U := Fin (Fintype.card S)) lx T)
      (r * Separation.gridGroupOrder (Fintype.card S)) := by
  exact finite_grid_separation_rankAtMost T lx ly
    (lt_of_lt_of_le (by decide : 0 < 2)
      (Separation.gridWidth_ge_two (Fintype.card S)))
    (Separation.grid_population_bound (Fintype.card S))
    (Separation.grid_population_bound (Fintype.card S)) hT hlabels

theorem auxiliary_separation_borderRankAtMost
    [Fintype X] [Fintype Y] [Fintype Z]
    (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
    (hT : Tensor.BorderRankAtMost T r)
    (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
    Tensor.BorderRankAtMost (separatedTensor (U := Fin (Fintype.card S)) lx T)
      (r * Separation.gridGroupOrder (Fintype.card S)) := by
  exact finite_grid_separation_borderRankAtMost T lx ly
    (lt_of_lt_of_le (by decide : 0 < 2)
      (Separation.gridWidth_ge_two (Fintype.card S)))
    (Separation.grid_population_bound (Fintype.card S))
    (Separation.grid_population_bound (Fintype.card S)) hT hlabels

theorem auxiliary_separation_directSum_rankAtMost
    (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
    (hT : Tensor.RankAtMost T r)
    (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
    Tensor.RankAtMost (Tensor.dependentDirectSum (fun s =>
      Tensor.product (labelFiberTensor lx ly T s)
        (Tensor.dotPairing (Fin (Fintype.card S)))))
      (r * Separation.gridGroupOrder (Fintype.card S)) := by
  exact separatedTensor_fiber_directSum_rankAtMost T lx ly hlabels
    (auxiliary_separation_rankAtMost T lx ly hT hlabels)

theorem auxiliary_separation_directSum_borderRankAtMost
    [Fintype X] [Fintype Y] [Fintype Z]
    (T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
    (hT : Tensor.BorderRankAtMost T r)
    (hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
    Tensor.BorderRankAtMost (Tensor.dependentDirectSum (fun s =>
      Tensor.product (labelFiberTensor lx ly T s)
        (Tensor.dotPairing (Fin (Fintype.card S)))))
      (r * Separation.gridGroupOrder (Fintype.card S)) := by
  exact separatedTensor_fiber_directSum_borderRankAtMost T lx ly hlabels
    (auxiliary_separation_borderRankAtMost T lx ly hT hlabels)

end AuxiliarySeparation

theorem auxiliary_separation_logarithmic_cost :
    Filter.Tendsto
      (fun K : ℕ => Real.log (Separation.gridGroupOrder K : ℝ) / Real.log (K : ℝ))
      Filter.atTop (nhds 1) :=
  Separation.tendsto_log_gridGroupOrder_div_log_population

end MatrixMultiplication.Foundation

noncomputable section

namespace MatrixMultiplication

open MatrixMultiplication.Foundation Filter
open scoped Topology BigOperators

namespace AuxiliarySeparation

variable {X Y Z Label : Type*} [Fintype Label] [DecidableEq Label]

theorem directSum_rankAtMost (T : Tensor ℂ X Y Z)
    (readX : X → Label) (readY : Y → Label) {R : ℕ}
    (hT : Tensor.RankAtMost T R)
    (readable : ∀ x y z, T x y z ≠ 0 → readX x = readY y) :
    Tensor.RankAtMost (Tensor.dependentDirectSum (fun label =>
      Tensor.product (labelFiberTensor readX readY T label)
        (Tensor.dotPairing (Fin (Fintype.card Label)))))
      (R * Separation.gridGroupOrder (Fintype.card Label)) :=
  auxiliary_separation_directSum_rankAtMost T readX readY hT readable

theorem directSum_borderRankAtMost [Fintype X] [Fintype Y] [Fintype Z]
    (T : Tensor ℂ X Y Z) (readX : X → Label) (readY : Y → Label) {R : ℕ}
    (hT : Tensor.BorderRankAtMost T R)
    (readable : ∀ x y z, T x y z ≠ 0 → readX x = readY y) :
    Tensor.BorderRankAtMost (Tensor.dependentDirectSum (fun label =>
      Tensor.product (labelFiberTensor readX readY T label)
        (Tensor.dotPairing (Fin (Fintype.card Label)))))
      (R * Separation.gridGroupOrder (Fintype.card Label)) :=
  auxiliary_separation_directSum_borderRankAtMost T readX readY hT readable

theorem logarithmic_cost :
    Tendsto (fun K : ℕ => Real.log (Separation.gridGroupOrder K : ℝ) /
      Real.log (K : ℝ)) atTop (𝓝 1) :=
  auxiliary_separation_logarithmic_cost

theorem cost_minus_population_tendsto_zero {population : ℕ → ℕ} {rate : ℝ}
    (positive : ∀ n, 0 < population n) (rate_nonneg : 0 ≤ rate)
    (population_rate : Tendsto
      (fun n => Real.log (population n : ℝ) / (n : ℝ)) atTop (𝓝 rate)) :
    Tendsto (fun n =>
      (Real.log (LabelHierarchySeparation.poolAuxiliaryCost (population n) : ℝ) -
        Real.log (population n : ℝ)) / (n : ℝ)) atTop (𝓝 0) := by
  have h := (LabelHierarchySeparation.tendsto_log_poolAuxiliaryCost_div_nat
    positive rate_nonneg population_rate).sub population_rate
  simpa only [sub_self, sub_div] using h

end AuxiliarySeparation
end MatrixMultiplication

end

end MatrixAllFields

end OAI