lean/OAI/LinearAlgebra/MatrixMultiplication/Entropy/ComplexFiniteEntropy.lean

import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring

namespace OAI

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

noncomputable section

namespace MatrixMultiplication.Foundation

open scoped BigOperators

def entropyTerm (p : ℝ) : ℝ := -p * Real.log p

def finiteEntropy {A : Type*} [Fintype A] (p : A → ℝ) : ℝ := ∑ a, entropyTerm (p a)

theorem finiteEntropy_equiv {A B : Type*} [Fintype A] [Fintype B]
    (e : A ≃ B) (p : B → ℝ) : finiteEntropy (fun a => p (e a)) = finiteEntropy p :=
  e.sum_comp (fun b => entropyTerm (p b))

@[simp] theorem entropyTerm_zero : entropyTerm 0 = 0 := by simp [entropyTerm]

theorem entropyTerm_mul (p q : ℝ) :
    entropyTerm (p * q) = q * entropyTerm p + p * entropyTerm q := by
  by_cases hp : p = 0
  · simp [hp]
  by_cases hq : q = 0
  · simp [hq]
  rw [entropyTerm, Real.log_mul hp hq]
  unfold entropyTerm
  ring

theorem finiteEntropy_chain {A B : Type*} [Fintype A] [Fintype B]
    (p : A → ℝ) (q : A → B → ℝ) (hq : ∀ a, ∑ b, q a b = 1) :
    finiteEntropy (fun ab : A × B => p ab.1 * q ab.1 ab.2) =
      finiteEntropy p + ∑ a, p a * finiteEntropy (q a) := by
  simp only [finiteEntropy, Fintype.sum_prod_type, entropyTerm_mul,
    Finset.sum_add_distrib, ← Finset.sum_mul, ← Finset.mul_sum, hq, one_mul]

structure FiniteLaw (A : Type*) [Fintype A] where
  mass : A → ℝ
  nonneg : ∀ a, 0 ≤ mass a
  total : ∑ a, mass a = 1

namespace FiniteLaw

variable {A B : Type*} [Fintype A] [Fintype B]

def map (p : FiniteLaw A) (f : A → B) : FiniteLaw B := by
  classical
  exact
    { mass := fun b => ∑ a, if f a = b then p.mass a else 0
      nonneg := fun b => Finset.sum_nonneg fun a _ => by
        split_ifs
        · exact p.nonneg a
        · exact le_rfl
      total := by
        rw [Finset.sum_comm]
        simpa using p.total }

@[simp] theorem map_mass [DecidableEq B] (p : FiniteLaw A) (f : A → B) (b : B) :
    (p.map f).mass b = ∑ a, if f a = b then p.mass a else 0 := by
  classical
  dsimp only [map]
  apply Finset.sum_congr rfl
  intro a _
  by_cases h : f a = b <;> simp only [h, ite_true, ite_false]

theorem map_mass_apply (p : FiniteLaw A) (f : A → B)
    (hf : Function.Injective f) (a : A) : (p.map f).mass (f a) = p.mass a := by
  classical
  simp [map_mass, hf.eq_iff]

theorem map_entropy_of_injective (p : FiniteLaw A) (f : A → B)
    (hf : Function.Injective f) : finiteEntropy (p.map f).mass = finiteEntropy p.mass := by
  classical
  have hterm (b : B) : entropyTerm ((p.map f).mass b) =
      ∑ a, if f a = b then entropyTerm (p.mass a) else 0 := by
    by_cases hb : ∃ a, f a = b
    · obtain ⟨a, rfl⟩ := hb
      rw [map_mass_apply p f hf]
      simp [hf.eq_iff]
    · simp [map_mass, not_exists.mp hb]
  unfold finiteEntropy
  simp_rw [hterm]
  rw [Finset.sum_comm]
  simp

def joint (p : FiniteLaw A) (q : A → FiniteLaw B) : FiniteLaw (A × B) where
  mass ab := p.mass ab.1 * (q ab.1).mass ab.2
  nonneg ab := mul_nonneg (p.nonneg ab.1) ((q ab.1).nonneg ab.2)
  total := by simp [Fintype.sum_prod_type, ← Finset.mul_sum, (q _).total, p.total]

theorem joint_entropy (p : FiniteLaw A) (q : A → FiniteLaw B) :
    finiteEntropy (p.joint q).mass =
      finiteEntropy p.mass + ∑ a, p.mass a * finiteEntropy (q a).mass :=
  finiteEntropy_chain p.mass (fun a => (q a).mass) (fun a => (q a).total)

theorem mass_le_map_mass (p : FiniteLaw A) (f : A → B) (a : A) :
    p.mass a ≤ (p.map f).mass (f a) := by
  classical
  rw [map_mass]
  have h := Finset.single_le_sum
    (s := (Finset.univ : Finset A))
    (f := fun a' => if f a' = f a then p.mass a' else 0)
    (fun a' _ => by
      split_ifs
      · exact p.nonneg a'
      · exact le_rfl)
    (Finset.mem_univ a)
  simpa using h

def conditional (p : FiniteLaw A) (f : A → B) (b : B) : FiniteLaw A := by
  classical
  exact if hzero : (p.map f).mass b = 0 then p else
    { mass := fun a => (if f a = b then p.mass a else 0) / (p.map f).mass b
      nonneg := fun a => div_nonneg
        (by split_ifs; exact p.nonneg a; exact le_rfl) ((p.map f).nonneg b)
      total := by
        rw [← Finset.sum_div, ← map_mass p f b, div_self hzero] }

theorem map_mass_mul_conditional [DecidableEq B]
    (p : FiniteLaw A) (f : A → B) (b : B) (a : A) :
    (p.map f).mass b * (p.conditional f b).mass a =
      if f a = b then p.mass a else 0 := by
  classical
  by_cases hzero : (p.map f).mass b = 0
  · have ha : f a = b → p.mass a = 0 := by
      intro hab
      have hle := p.mass_le_map_mass f a
      rw [hab, hzero] at hle
      exact le_antisymm hle (p.nonneg a)
    rw [hzero, zero_mul]
    by_cases hab : f a = b
    · rw [ite_eq_left hab, ha hab]
    · rw [ite_eq_right hab]
  · simp only [conditional, hzero, ↓reduceDIte]
    by_cases hab : f a = b
    · simpa only [hab, ite_true] using (mul_div_cancel₀ (p.mass a) hzero)
    · simp only [hab, ite_false, zero_div, mul_zero]

theorem entropy_eq_map_add_conditional (p : FiniteLaw A) (f : A → B) :
    finiteEntropy p.mass = finiteEntropy (p.map f).mass +
      ∑ b, (p.map f).mass b * finiteEntropy (p.conditional f b).mass := by
  classical
  let graph : A → B × A := fun a => (f a, a)
  have hinj : Function.Injective graph := fun _ _ h => congrArg Prod.snd h
  have hmass : ((p.map f).joint (p.conditional f)).mass = (p.map graph).mass := by
    funext ba
    rcases ba with ⟨b, a⟩
    change (p.map f).mass b * (p.conditional f b).mass a = (p.map graph).mass (b, a)
    rw [map_mass_mul_conditional, map_mass, Finset.sum_eq_single a]
    · simp only [graph, Prod.mk.injEq, and_true]
    · intro x _ hxa
      simp only [graph, Prod.mk.injEq, hxa, and_false, ite_false]
    · intro ha
      exact (ha (Finset.mem_univ a)).elim
  calc
    finiteEntropy p.mass = finiteEntropy (p.map graph).mass :=
      (p.map_entropy_of_injective graph hinj).symm
    _ = finiteEntropy ((p.map f).joint (p.conditional f)).mass := by rw [hmass]
    _ = _ := (p.map f).joint_entropy (p.conditional f)

end FiniteLaw
end MatrixMultiplication.Foundation

end

end OAI