lean/OAI/Combinatorics/MatchingEntropy/Simplex.lean

import OAI.Combinatorics.MatchingEntropy.Model

namespace OAI

noncomputable section

open scoped BigOperators

universe uIndex

namespace MatchingEntropyBounds

theorem probabilitySimplex_eq_range (Index : Type uIndex) [Fintype Index] :
    probabilitySimplex Index =
      Set.range (fun simplex : Convexity.StdSimplex ℝ Index ↦ ⇑simplex.weights) := by
  rw [Convexity.StdSimplex.range_toFun_comp_weights]
  ext weights
  simp [probabilitySimplex]

theorem convex_probabilitySimplex (Index : Type uIndex) [Fintype Index] :
    Convex ℝ (probabilitySimplex Index) := by
  intro leftWeights leftMem rightWeights rightMem leftScale rightScale
    leftNonneg rightNonneg total
  constructor
  · intro index
    exact add_nonneg (mul_nonneg leftNonneg (leftMem.1 index))
      (mul_nonneg rightNonneg (rightMem.1 index))
  · simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul, Finset.sum_add_distrib,
      ← Finset.mul_sum, leftMem.2, rightMem.2, mul_one, total]

theorem single_mem_probabilitySimplex {Index : Type uIndex} [Fintype Index]
    [DecidableEq Index] (index : Index) : Pi.single index 1 ∈ probabilitySimplex Index := by
  constructor
  · intro other
    simp only [Pi.single_apply]
    split <;> norm_num
  · simp

theorem mem_Icc_of_mem_probabilitySimplex {Index : Type uIndex} [Fintype Index]
    {weights : Index → ℝ} (mem : weights ∈ probabilitySimplex Index) (index : Index) :
    weights index ∈ Set.Icc (0 : ℝ) 1 := by
  refine ⟨mem.1 index, ?_⟩
  rw [← mem.2]
  exact Finset.single_le_sum (fun other _ ↦ mem.1 other) (Finset.mem_univ index)

theorem isCompact_probabilitySimplex (Index : Type uIndex) [Fintype Index] :
    IsCompact (probabilitySimplex Index) := by
  rw [probabilitySimplex_eq_range]
  exact isCompact_range
    (Convexity.StdSimplex.isEmbedding_toFun_comp_weights ℝ Index).continuous

end MatchingEntropyBounds

end

end OAI