/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Data.Fintype.CardEmbedding
import Mathlib.Probability.UniformOn
import Mathlib.Probability.Notation

/-!
# Birthday Problem

This file proves Theorem 93 from the [100 Theorems List](https://www.cs.ru.nl/~freek/100/).

As opposed to the standard probabilistic statement, we instead state the birthday problem
in terms of injective functions. The general result about `Fintype.card (α ↪ β)` which this proof
uses is `Fintype.card_embedding_eq`.
-/


namespace Theorems100

local notation "|" x "|" => Finset.card x

local notation "‖" x "‖" => Fintype.card x

/-- **Birthday Problem**: set cardinality interpretation. -/
theorem birthday :
    2 * ‖Fin 23 ↪ Fin 365‖ < ‖Fin 23 → Fin 365‖ ∧ 2 * ‖Fin 22 ↪ Fin 365‖ > ‖Fin 22 → Fin 365‖ := by sorry
section MeasureTheory

open MeasureTheory ProbabilityTheory

open scoped ProbabilityTheory ENNReal

variable {n m : ℕ}

/- In order for Lean to understand that we can take probabilities in `Fin 23 → Fin 365`, we must
tell Lean that there is a `MeasurableSpace` structure on the space. Note that this instance
is only for `Fin m` - Lean automatically figures out that the function space `Fin n → Fin m`
is _also_ measurable, by using `MeasurableSpace.pi`, and furthermore that all sets are measurable,
from `MeasurableSingletonClass.pi`. -/
instance : MeasurableSpace (Fin m) :=
  ⊤

instance : MeasurableSingletonClass (Fin m) :=
  ⟨fun _ => trivial⟩

/- We then endow the space with a canonical measure, which is called ℙ.
We define this to be the conditional counting measure. -/
noncomputable instance measureSpace : MeasureSpace (Fin n → Fin m) :=
  ⟨uniformOn Set.univ⟩

theorem FinFin.measure_apply {s : Set <| Fin n → Fin m} :
    ℙ s = |s.toFinite.toFinset| / ‖Fin n → Fin m‖ := by sorry
theorem birthday_measure :
    ℙ ({f | (Function.Injective f)} : Set ((Fin 23) → (Fin 365))) < 1 / 2 := by sorry
end MeasureTheory

end Theorems100