lean/OAI/Combinatorics/ExpectationThreshold/ProbabilityFinite.lean

import Mathlib.Basic.Real.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Pi
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.SplitIfs

namespace OAI

/-!
# Finite probability laws

Probabilities and expectations are finite real sums. Independent samples,
product laws, and uniform laws satisfy the usual elementary inequalities.
-/

open scoped BigOperators
noncomputable section
attribute [local instance] Classical.propDecidable

namespace TalagrandThreshold

structure FiniteLaw (Ω : Type*) [Fintype Ω] where
  weight : Ω → ℝ
  nonneg : ∀ ω, 0 ≤ weight ω
  total : ∑ ω, weight ω = 1

namespace FiniteLaw

def expectation {Ω : Type*} [Fintype Ω] (μ : FiniteLaw Ω) (f : Ω → ℝ) : ℝ :=
  ∑ ω, μ.weight ω * f ω

def prob {Ω : Type*} [Fintype Ω] (μ : FiniteLaw Ω) (P : Ω → Prop) : ℝ := by
  classical
  exact ∑ ω, if P ω then μ.weight ω else 0

def IsPositive {Ω : Type*} [Fintype Ω] (μ : FiniteLaw Ω) : Prop :=
  ∀ ω, 0 < μ.weight ω

variable {Ω Γ : Type*} [Fintype Ω] [Fintype Γ]

@[simp] theorem expectation_const (μ : FiniteLaw Ω) (c : ℝ) :
    μ.expectation (fun _ => c) = c := by
  rw [expectation, ← Finset.sum_mul, μ.total, one_mul]

theorem expectation_add (μ : FiniteLaw Ω) (f g : Ω → ℝ) :
    μ.expectation (fun ω => f ω + g ω) = μ.expectation f + μ.expectation g := by
  simp [expectation, mul_add, Finset.sum_add_distrib]

theorem expectation_sub (μ : FiniteLaw Ω) (f g : Ω → ℝ) :
    μ.expectation (fun ω => f ω - g ω) = μ.expectation f - μ.expectation g := by
  simp [expectation, mul_sub, Finset.sum_sub_distrib]

theorem expectation_smul (μ : FiniteLaw Ω) (c : ℝ) (f : Ω → ℝ) :
    μ.expectation (fun ω => c * f ω) = c * μ.expectation f := by
  simp [expectation, Finset.mul_sum, mul_left_comm]

theorem expectation_sum {ι : Type*} [Fintype ι] (μ : FiniteLaw Ω) (f : ι → Ω → ℝ) :
    μ.expectation (fun ω => ∑ i, f i ω) = ∑ i, μ.expectation (f i) := by
  simp only [expectation, Finset.mul_sum]
  exact Finset.sum_comm

theorem expectation_nonneg (μ : FiniteLaw Ω) {f : Ω → ℝ} (hf : ∀ ω, 0 ≤ f ω) :
    0 ≤ μ.expectation f :=
  Finset.sum_nonneg fun ω _ => mul_nonneg (μ.nonneg ω) (hf ω)

theorem expectation_mono (μ : FiniteLaw Ω) {f g : Ω → ℝ} (hfg : ∀ ω, f ω ≤ g ω) :
    μ.expectation f ≤ μ.expectation g :=
  Finset.sum_le_sum fun ω _ => mul_le_mul_of_nonneg_left (hfg ω) (μ.nonneg ω)

theorem prob_eq_expectation (μ : FiniteLaw Ω) (P : Ω → Prop) :
    μ.prob P = μ.expectation (fun ω => if P ω then 1 else 0) := by
  classical
  simp [prob, expectation, mul_ite]

@[simp] theorem prob_true (μ : FiniteLaw Ω) : μ.prob (fun _ => True) = 1 := by
  simp [prob, μ.total]

@[simp] theorem prob_false (μ : FiniteLaw Ω) : μ.prob (fun _ => False) = 0 := by
  simp [prob]

theorem prob_nonneg (μ : FiniteLaw Ω) (P : Ω → Prop) : 0 ≤ μ.prob P := by
  classical
  exact Finset.sum_nonneg fun ω _ => by split_ifs <;> simp [μ.nonneg]

theorem prob_mono (μ : FiniteLaw Ω) {P Q : Ω → Prop} (hPQ : ∀ ω, P ω → Q ω) :
    μ.prob P ≤ μ.prob Q := by
  classical
  apply Finset.sum_le_sum
  intro ω _
  by_cases hp : P ω
  · simp [hp, hPQ ω hp]
  · by_cases hq : Q ω <;> simp [hp, hq, μ.nonneg ω]

theorem prob_le_one (μ : FiniteLaw Ω) (P : Ω → Prop) : μ.prob P ≤ 1 := by
  simpa using μ.prob_mono (Q := fun _ => True) (fun _ _ => trivial)

theorem prob_compl (μ : FiniteLaw Ω) (P : Ω → Prop) :
    μ.prob (fun ω => ¬ P ω) = 1 - μ.prob P := by
  classical
  have h : μ.prob P + μ.prob (fun ω => ¬ P ω) = 1 := by
    rw [prob, prob, ← Finset.sum_add_distrib, ← μ.total]
    apply Finset.sum_congr rfl
    intro ω _
    by_cases hp : P ω <;> simp [hp]
  linarith

theorem prob_union_le (μ : FiniteLaw Ω) (P Q : Ω → Prop) :
    μ.prob (fun ω => P ω ∨ Q ω) ≤ μ.prob P + μ.prob Q := by
  classical
  rw [prob, prob, prob, ← Finset.sum_add_distrib]
  apply Finset.sum_le_sum
  intro ω _
  by_cases hp : P ω <;> by_cases hq : Q ω <;> simp [hp, hq, μ.nonneg]

theorem prob_le_expectation (μ : FiniteLaw Ω) (P : Ω → Prop) (f : Ω → ℝ)
    (hf : ∀ ω, 0 ≤ f ω) (hP : ∀ ω, P ω → 1 ≤ f ω) : μ.prob P ≤ μ.expectation f := by
  classical
  rw [prob_eq_expectation]
  apply μ.expectation_mono
  intro ω
  by_cases hp : P ω
  · simpa [hp] using hP ω hp
  · simpa [hp] using hf ω

theorem prob_ge_le_expectation_div (μ : FiniteLaw Ω) (f : Ω → ℝ)
    (hf : ∀ ω, 0 ≤ f ω) {t : ℝ} (ht : 0 < t) :
    μ.prob (fun ω => t ≤ f ω) ≤ μ.expectation f / t := by
  have h := μ.prob_le_expectation (fun ω => t ≤ f ω) (fun ω => t⁻¹ * f ω)
    (fun ω => mul_nonneg (inv_nonneg.mpr ht.le) (hf ω))
    (fun ω hω => by simpa [ht.ne'] using mul_le_mul_of_nonneg_left hω (inv_nonneg.mpr ht.le))
  rw [expectation_smul] at h
  simpa [div_eq_mul_inv, mul_comm] using h

def prod (μ : FiniteLaw Ω) (ν : FiniteLaw Γ) : FiniteLaw (Ω × Γ) where
  weight ω := μ.weight ω.1 * ν.weight ω.2
  nonneg ω := mul_nonneg (μ.nonneg _) (ν.nonneg _)
  total := by
    simp only [Fintype.sum_prod_type, ← Finset.mul_sum, ν.total, mul_one, μ.total]

@[simp] theorem prod_weight (μ : FiniteLaw Ω) (ν : FiniteLaw Γ) (ω : Ω) (γ : Γ) :
    (μ.prod ν).weight (ω, γ) = μ.weight ω * ν.weight γ := rfl

theorem expectation_prod (μ : FiniteLaw Ω) (ν : FiniteLaw Γ) (f : Ω × Γ → ℝ) :
    (μ.prod ν).expectation f = μ.expectation (fun ω => ν.expectation (fun γ => f (ω, γ))) := by
  simp [expectation, prod, Fintype.sum_prod_type, Finset.mul_sum, mul_assoc]

theorem prod_isPositive (μ : FiniteLaw Ω) (ν : FiniteLaw Γ)
    (hμ : μ.IsPositive) (hν : ν.IsPositive) : (μ.prod ν).IsPositive :=
  fun outcome => mul_pos (hμ outcome.1) (hν outcome.2)

def pi {ι : Type*} [Fintype ι] [DecidableEq ι] {A : ι → Type*} [∀ i, Fintype (A i)]
    (μ : ∀ i, FiniteLaw (A i)) : FiniteLaw (∀ i, A i) := by
  classical
  exact {
    weight := fun ω => ∏ i, (μ i).weight (ω i)
    nonneg := fun ω => Finset.prod_nonneg fun i _ => (μ i).nonneg (ω i)
    total := by
      rw [← Fintype.prod_sum (fun i a => (μ i).weight a)]
      simp_rw [(μ _).total]
      simp }

@[simp] theorem pi_weight {ι : Type*} [Fintype ι] [DecidableEq ι] {A : ι → Type*} [∀ i, Fintype (A i)]
    (μ : ∀ i, FiniteLaw (A i)) (ω : ∀ i, A i) :
    (pi μ).weight ω = ∏ i, (μ i).weight (ω i) := rfl

def iid (μ : FiniteLaw Ω) (n : ℕ) : FiniteLaw (Fin n → Ω) := pi (fun _ : Fin n => μ)

@[simp] theorem iid_weight (μ : FiniteLaw Ω) (n : ℕ) (ω : Fin n → Ω) :
    (iid μ n).weight ω = ∏ i, μ.weight (ω i) := rfl

def uniform (Ω : Type*) [Fintype Ω] [Nonempty Ω] : FiniteLaw Ω where
  weight _ := (Fintype.card Ω : ℝ)⁻¹
  nonneg _ := inv_nonneg.mpr (Nat.cast_nonneg _)
  total := by
    have hN : (Fintype.card Ω : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr Fintype.card_ne_zero
    simp [hN]

theorem uniform_isPositive (Ω : Type*) [Fintype Ω] [Nonempty Ω] :
    (uniform Ω).IsPositive := fun _ => inv_pos.mpr (Nat.cast_pos.mpr Fintype.card_pos)

theorem uniform_prob (Ω : Type*) [Fintype Ω] [Nonempty Ω] (P : Ω → Prop) :
    (uniform Ω).prob P = ((Finset.univ.filter P).card : ℝ) / Fintype.card Ω := by
  simp [prob, uniform, ← Finset.sum_filter, div_eq_mul_inv]

end FiniteLaw

end TalagrandThreshold

end

end OAI