| import OAI.Combinatorics.MatchingEntropy.Components |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators Topology |
| open Filter |
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| universe uV uE uI uJ uA uB uC uW uF uD uι |
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| namespace MatchingEntropyBounds.Refined |
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| variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
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| theorem empty_conventions (G : LooplessGraph V E) (hV : Fintype.card V = 0) : |
| Fintype.card E = 0 ∧ Fintype.card G.Matching = 1 ∧ G.polytope = {0} ∧ |
| (∀ x ∈ G.polytope, entropy x = 0 ∧ complementEntropy x = 0 ∧ maxMatchingEntropy G x = 0) ∧ |
| (∀ β : E → ℝ, weightedPartition G β = 1 ∧ weightedObjective β 0 = 0) ∧ |
| (∀ k : ℕ, 1 ≤ k → (Real.exp (-2) * (k : ℝ))^0 = 1) := by |
| have : IsEmpty V := Fintype.card_eq_zero_iff.mp hV |
| have : IsEmpty E := ⟨fun e => isEmptyElim (G.left e)⟩ |
| let M : G.Matching := ⟨∅,fun v => isEmptyElim v⟩ |
| have : Nonempty G.Matching := ⟨M⟩ |
| have : Subsingleton G.Matching := inferInstance |
| have hc : Fintype.card G.Matching = 1 := by |
| let : Unique G.Matching := ⟨⟨M⟩, fun _ => Subsingleton.elim _ _⟩ |
| exact Fintype.card_unique |
| have hi : G.indicator M = 0 := Subsingleton.elim _ _ |
| have hp : G.polytope = {0} := by |
| apply Set.Subset.antisymm |
| · intro x _ |
| exact Set.mem_singleton_iff.mpr (Subsingleton.elim _ _) |
| · rintro x rfl |
| rw [← hi] |
| exact subset_convexHull ℝ _ (Set.mem_range_self M) |
| refine ⟨Fintype.card_eq_zero, hc, hp, ?_, ?_, fun _ _ => pow_zero _⟩ |
| · intro x hx |
| have hF : entropy x = 0 := by simp [entropy] |
| have hB : complementEntropy x = 0 := by simp [complementEntropy, entropy] |
| refine ⟨hF,hB,le_antisymm ?_ (maxMatchingEntropy_nonneg G hx)⟩ |
| simpa [hF] using refined_upper G x hx |
| · intro β |
| constructor |
| · simp [weightedPartition, hc] |
| · simp [weightedObjective, entropy] |
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| theorem no_matching_polytope {V : Type uV} {E : Type uE} |
| [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) : |
| IsEmpty G.Matching ↔ G.polytope = ∅ := by |
| constructor |
| · intro hm |
| have : IsEmpty G.Matching := hm |
| exact Set.eq_empty_iff_forall_notMem.mpr (fun x hx => by |
| obtain ⟨M⟩ := matching_nonempty_of_mem G hx |
| exact isEmptyElim M) |
| · intro hp |
| exact ⟨fun M => by |
| have h := subset_convexHull ℝ (Set.range G.indicator) (Set.mem_range_self M) |
| change G.indicator M ∈ G.polytope at h |
| rw [hp] at h |
| exact h⟩ |
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| theorem zero_selected_bound {V : Type uV} {E : Type uE} |
| [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (m : ℕ) (hm : 0 < m) : |
| (Real.exp (-2) * (0 : ℝ))^m = 0 ∧ (0 : ℝ) ≤ Fintype.card G.Matching := by |
| simp [zero_pow (Nat.ne_of_gt hm)] |
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| theorem zero_regular {V : Type uV} {E : Type uE} |
| [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (hV : 0 < Fintype.card V) |
| (hreg : ∀ v, (Finset.univ.filter (G.Incident v)).card = 0) : |
| Fintype.card E = 0 ∧ IsEmpty G.Matching := by |
| have he : IsEmpty E := by |
| refine ⟨fun e => ?_⟩ |
| have hh := Finset.card_eq_zero.mp (hreg (G.left e)) |
| have hm : e ∈ Finset.univ.filter (G.Incident (G.left e)) := by |
| simp [LooplessGraph.Incident] |
| rw [hh] at hm |
| exact Finset.notMem_empty e hm |
| have : IsEmpty E := he |
| obtain ⟨v⟩ := Fintype.card_pos_iff.mp hV |
| exact ⟨Fintype.card_eq_zero, ⟨fun M => by obtain ⟨e,_,_⟩ := M.property v; exact isEmptyElim e⟩⟩ |
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| theorem one_selected_bound {V : Type uV} {E : Type uE} |
| [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (M : G.Matching) : |
| 1 ≤ Fintype.card G.Matching := Fintype.card_pos_iff.mpr ⟨M⟩ |
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| theorem one_partition_exact {V : Type uV} {E : Type uE} |
| [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] (G : LooplessGraph V E) (M : G.Matching) |
| (hcover : M.val = Finset.univ) : Fintype.card G.Matching = 1 := by |
| have hu : ∀ N : G.Matching, N = M := by |
| intro N |
| apply Subtype.ext |
| apply Finset.Subset.antisymm |
| · rw [hcover] |
| exact Finset.subset_univ _ |
| · intro e he |
| obtain ⟨f,hf,_⟩ := N.property (G.left e) |
| have hfm : f ∈ M.val := hcover ▸ Finset.mem_univ f |
| have hfe := (M.property (G.left e)).unique ⟨hfm,hf.2⟩ ⟨he,Or.inl rfl⟩ |
| exact hfe ▸ hf.1 |
| have : Subsingleton G.Matching := ⟨fun a b => (hu a).trans (hu b).symm⟩ |
| exact by |
| let : Unique G.Matching := ⟨⟨M⟩, fun _ => Subsingleton.elim _ _⟩ |
| exact Fintype.card_unique |
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| theorem pointwise_endpoints (G : LooplessGraph V E) (m : ℕ) |
| (hm : 0 < m) (hV : Fintype.card V = 2 * m) {x : E → ℝ} (hx : x ∈ G.polytope) : |
| entropy x - 2 * complementEntropy x ≤ maxMatchingEntropy G x ∧ |
| (m = 1 → maxMatchingEntropy G x = entropy x) := by |
| have hl := refined_lower G m hm hV x hx |
| change entropy x - (2 - 2 / (m : ℝ)) * complementEntropy x ≤ maxMatchingEntropy G x at hl |
| have hB := B_nonneg G hx |
| have hmR : (0 : ℝ) < m := by exact_mod_cast hm |
| constructor |
| · have hpos := mul_nonneg (div_nonneg (by norm_num : (0:ℝ) ≤ 2) hmR.le) hB |
| nlinarith |
| · intro hh |
| subst m |
| norm_num at hl |
| exact le_antisymm (refined_upper G x hx) hl |
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| end MatchingEntropyBounds.Refined |
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| end |
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| end OAI |
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