lean/OAI/Combinatorics/MatchingEntropy/Model.lean

import Mathlib

namespace OAI

noncomputable section

open scoped BigOperators Topology
open Filter

universe uV uE uIndex uι

namespace MatchingEntropyBounds

structure LooplessGraph (V : Type uV) (E : Type uE) where
  left : E → V
  right : E → V
  loopless : ∀ e, left e ≠ right e

variable {V : Type uV} {E : Type uE}
  [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]

namespace LooplessGraph

def Incident (G : LooplessGraph V E) (v : V) (e : E) : Prop :=
  G.left e = v ∨ G.right e = v

instance incidentDecidable (G : LooplessGraph V E) (v : V) (e : E) :
    Decidable (G.Incident v e) := inferInstanceAs (Decidable (G.left e = v ∨ G.right e = v))

def IsPerfectMatching (G : LooplessGraph V E) (M : Finset E) : Prop :=
  ∀ v, ∃! e, e ∈ M ∧ G.Incident v e

abbrev Matching (G : LooplessGraph V E) := {M : Finset E // G.IsPerfectMatching M}

instance matchingFintype (G : LooplessGraph V E) : Fintype G.Matching :=
  Fintype.ofFinite _

def indicator (G : LooplessGraph V E) (M : G.Matching) (e : E) : ℝ :=
  if e ∈ M.val then 1 else 0

def polytope (G : LooplessGraph V E) : Set (E → ℝ) :=
  convexHull ℝ (Set.range G.indicator)

def mean (G : LooplessGraph V E) (p : G.Matching → ℝ) (e : E) : ℝ :=
  ∑ M, p M * G.indicator M e

end LooplessGraph

def probabilitySimplex (Index : Type uIndex) [Fintype Index] : Set (Index → ℝ) :=
  {weights | (∀ index, 0 ≤ weights index) ∧ ∑ index, weights index = 1}

namespace LooplessGraph

def feasibleLaws (G : LooplessGraph V E) (y : E → ℝ) : Set (G.Matching → ℝ) :=
  {p | p ∈ probabilitySimplex G.Matching ∧ G.mean p = y}

end LooplessGraph

def entropy {ι : Type uι} [Fintype ι] (p : ι → ℝ) : ℝ :=
  ∑ i, Real.negMulLog (p i)

def maxMatchingEntropy (G : LooplessGraph V E) (y : E → ℝ) : ℝ :=
  sSup (entropy '' G.feasibleLaws y)

namespace Refined

def complementEntropy (x : E → ℝ) : ℝ := entropy (fun e => 1 - x e)

end Refined

end MatchingEntropyBounds

end

end OAI