| import OAI.Combinatorics.MatchingEntropy.Counting |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators Topology |
| open Filter Module |
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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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| section SharedEquation |
| variable {A : Type uA} {B : Type uB} {C : Type uC} [AddCommGroup A] [AddCommGroup B] [AddCommGroup C] |
| [Module ℝ A] [Module ℝ B] [Module ℝ C] |
| [FiniteDimensional ℝ A] [FiniteDimensional ℝ B] [FiniteDimensional ℝ C] |
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| theorem quotient_finrank_shared (U : Submodule ℝ A) (W : Submodule ℝ B) (Z : Submodule ℝ C) |
| (f : A →ₗ[ℝ] B) (g : A →ₗ[ℝ] C) |
| (h : ∀ u, f u ∈ W → g u ∈ Z → u ∈ U) |
| (l : B →ₗ[ℝ] ℝ) (r : C →ₗ[ℝ] ℝ) (hl : W ≤ l.ker) (hr : Z ≤ r.ker) |
| (heq : ∀ u, l (f u) = r (g u)) (b : B) (hb : l b ≠ 0) : |
| finrank ℝ (A ⧸ U) + 1 ≤ finrank ℝ (B ⧸ W) + finrank ℝ (C ⧸ Z) := by |
| let F := (W.mkQ.comp f).prod (Z.mkQ.comp g) |
| let L : ((B ⧸ W) × (C ⧸ Z)) →ₗ[ℝ] ℝ := |
| (W.liftQ l hl).comp (LinearMap.fst ℝ _ _) - |
| (Z.liftQ r hr).comp (LinearMap.snd ℝ _ _) |
| have hk : F.ker ≤ U := by |
| intro u hu |
| have hh : (W.mkQ (f u), Z.mkQ (g u)) = 0 := hu |
| apply h u |
| · simpa using congrArg Prod.fst hh |
| · simpa using congrArg Prod.snd hh |
| have hL : ∀ u, F u ∈ L.ker := by |
| intro u |
| change l (f u) - r (g u) = 0 |
| exact sub_eq_zero.mpr (heq u) |
| let F' := F.codRestrict L.ker hL |
| have hk' : F'.ker ≤ U := by |
| intro u hu |
| apply hk |
| have hh := congrArg Subtype.val (show F' u = 0 from hu) |
| exact hh |
| have hdim := quotient_finrank_le_of_ker_le U F' hk' |
| have hsum := L.finrank_range_add_finrank_ker |
| have hpos : 1 ≤ finrank ℝ L.range := by |
| apply Submodule.one_le_finrank_iff.mpr |
| intro hh |
| have hm : L (W.mkQ b, 0) ∈ L.range := ⟨_, rfl⟩ |
| rw [hh] at hm |
| have hz : L (W.mkQ b, 0) = 0 := hm |
| apply hb |
| simpa [L] using hz |
| rw [Module.finrank_prod] at hsum |
| omega |
| end SharedEquation |
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| variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
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| def lowerObjective (c : ℝ) (x : E → ℝ) : ℝ := entropy x - c * complementEntropy x |
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| open LooplessGraph |
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| omit [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] in |
| theorem positiveExtension_self {x : E → ℝ} (hx : ∀ e, 0 ≤ x e) : |
| positiveExtension x (fun e => x e.val) = x := by |
| funext e |
| by_cases he : 0 < x e |
| · simp [positiveExtension, he] |
| · simp [positiveExtension, le_antisymm (le_of_not_gt he) (hx e)] |
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| def boundaryLinear (G : LooplessGraph V E) (x : E → ℝ) (S : Finset V) : |
| ({e // 0 < G.sideMarginal x S e} → ℝ) →ₗ[ℝ] ℝ := |
| (LinearMap.proj none).comp |
| ((G.onSide S).degreeLinear.comp (positiveExtension (G.sideMarginal x S))) |
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| omit [Fintype V] [DecidableEq E] in |
| theorem boundaryLinear_restriction (G : LooplessGraph V E) (x : E → ℝ) (S : Finset V) |
| (u : {e // 0 < x e} → ℝ) : |
| boundaryLinear G x S (G.sideRestriction x S u) = G.cutMass (positiveExtension x u) S := by |
| change (G.onSide S).degree (fun e => positiveExtension x u e.val) none = _ |
| exact G.onSide_degree_none S _ |
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| omit [Fintype V] [DecidableEq E] in |
| theorem boundaryLinear_face (G : LooplessGraph V E) (x : E → ℝ) (S : Finset V) : |
| (G.onSide S).faceSpace (G.sideMarginal x S) ≤ (boundaryLinear G x S).ker := by |
| intro u hu |
| exact congrFun hu.1 none |
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| theorem faceCodimension_le_sides_shared (G : LooplessGraph V E) {x : E → ℝ} |
| (heven : Even (Fintype.card V)) (hx : x ∈ G.constraintPolytope) |
| (S : Finset V) (hS : Odd S.card) (ht : G.cutMass x S = 1) : |
| G.faceCodimension x + 1 ≤ (G.onSide S).faceCodimension (G.sideMarginal x S) + |
| (G.onSide Sᶜ).faceCodimension (G.sideMarginal x Sᶜ) := by |
| classical |
| apply quotient_finrank_shared (G.faceSpace x) |
| ((G.onSide S).faceSpace (G.sideMarginal x S)) |
| ((G.onSide Sᶜ).faceSpace (G.sideMarginal x Sᶜ)) |
| (G.sideRestriction x S) (G.sideRestriction x Sᶜ) |
| (fun _ => G.faceSpace_of_side_spaces heven hx S hS ht) |
| (boundaryLinear G x S) (boundaryLinear G x Sᶜ) |
| (boundaryLinear_face G x S) (boundaryLinear_face G x Sᶜ) |
| (fun u => by simp only [boundaryLinear_restriction, cutMass, cut_compl]) |
| (fun e => G.sideMarginal x S e.val) |
| have hext := positiveExtension_self (x := G.sideMarginal x S) (fun e => hx.1 e.val) |
| change (G.onSide S).degree |
| (positiveExtension (G.sideMarginal x S) (fun e => G.sideMarginal x S e.val)) none ≠ 0 |
| rw [hext, G.onSide_degree_none, ht] |
| norm_num |
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| omit [Fintype E] [DecidableEq E] in |
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| theorem incident_of_card_two (G : LooplessGraph V E) (hV : Fintype.card V = 2) |
| (v : V) (e : E) : G.Incident v e := by |
| have hp : ({G.left e, G.right e} : Finset V) = Finset.univ := by |
| apply Finset.eq_of_subset_of_card_le (Finset.subset_univ _) |
| simp [hV, G.loopless e] |
| have hv : v ∈ ({G.left e, G.right e} : Finset V) := hp ▸ Finset.mem_univ v |
| simpa only [Finset.mem_insert, Finset.mem_singleton, eq_comm, Incident] using hv |
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| omit [DecidableEq E] in |
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| theorem faceCodimension_two (G : LooplessGraph V E) (hV : Fintype.card V = 2) |
| (x : E → ℝ) : G.faceCodimension x ≤ 1 := by |
| classical |
| let L : ({e // 0 < x e} → ℝ) →ₗ[ℝ] ℝ := |
| (Finset.univ.sum (fun e : E => LinearMap.proj e)).comp (positiveExtension x) |
| have heven : Even (Fintype.card V) := by simp [hV] |
| have hnt : ¬ ∃ S : Finset V, Odd S.card ∧ 3 ≤ S.card ∧ 3 ≤ Sᶜ.card ∧ G.cutMass x S = 1 := by |
| rintro ⟨S, _, hlo, _, _⟩ |
| have hc := Finset.card_le_univ S |
| omega |
| have hker : L.ker ≤ G.faceSpace x := by |
| intro u hu |
| have hz : ∑ e, positiveExtension x u e = 0 := by simpa [L] using hu |
| have hd : ∀ v, G.degree (positiveExtension x u) v = 0 := by |
| intro v |
| simpa only [degree, incident_of_card_two G hV v, ite_true] using hz |
| refine ⟨funext hd, ?_⟩ |
| exact G.trivial_tight_cut_preserved heven hnt _ hd |
| simpa [faceCodimension] using quotient_finrank_le_of_ker_le (G.faceSpace x) L hker |
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| theorem sharp_face_rank_aux (n : ℕ) : |
| ∀ {W : Type uW} {F : Type uF} [Fintype W] [Fintype F] [DecidableEq W] [DecidableEq F] |
| (G : LooplessGraph W F), Fintype.card W=n → Even n → 4≤n → |
| ∀ x ∈ G.constraintPolytope, 2 * (G.faceCodimension x + 2) ≤ 3 * n := by |
| induction n using Nat.strong_induction_on with |
| | h n ih => |
| intro W F _ _ _ _ G hcard heven hfour x hx |
| have heven' : Even (Fintype.card W) := hcard ▸ heven |
| by_cases hcut : ∃ S : Finset W, Odd S.card ∧ 3≤S.card ∧ 3≤Sᶜ.card ∧ G.cutMass x S=1 |
| · obtain ⟨S,hS,hlo,hhi,ht⟩ := hcut |
| have hc := odd_compl_of_even heven' hS |
| have hlt1 : Fintype.card (Option {v // v ∈ S}) < n := by |
| simp only [Fintype.card_option,Fintype.card_coe] |
| have hco := Finset.card_compl S |
| omega |
| have hlt2 : Fintype.card (Option {v // v ∈ Sᶜ}) < n := by |
| simp only [Fintype.card_option,Fintype.card_coe] |
| have hco := Finset.card_compl S |
| have hle := Finset.card_le_univ S |
| omega |
| have hv1 : Even (Fintype.card (Option {v // v ∈ S})) := by |
| simpa only [Fintype.card_option,Fintype.card_coe] using hS.add_odd odd_one |
| have hv2 : Even (Fintype.card (Option {v // v ∈ Sᶜ})) := by |
| simpa only [Fintype.card_option,Fintype.card_coe] using hc.add_odd odd_one |
| have hn1 : 4≤Fintype.card (Option {v // v ∈ S}) := by |
| simp only [Fintype.card_option,Fintype.card_coe]; omega |
| have hn2 : 4≤Fintype.card (Option {v // v ∈ Sᶜ}) := by |
| simp only [Fintype.card_option,Fintype.card_coe]; omega |
| have hx1 := G.restriction_mem_constraints hx S hS ht |
| have htc : G.cutMass x Sᶜ=1 := by simpa only [cutMass,cut_compl] using ht |
| have hx2 := G.restriction_mem_constraints hx Sᶜ hc htc |
| have hb1 := ih _ hlt1 (G.onSide S) rfl hv1 hn1 _ hx1 |
| have hb2 := ih _ hlt2 (G.onSide Sᶜ) rfl hv2 hn2 _ hx2 |
| have hb := faceCodimension_le_sides_shared G heven' hx S hS ht |
| have hsum : Fintype.card (Option {v // v ∈ S}) + Fintype.card (Option {v // v ∈ Sᶜ})=n+2 := by |
| simp only [Fintype.card_option,Fintype.card_coe] |
| have hh := Finset.card_compl_add_card S |
| omega |
| change 2 * ((G.onSide S).faceCodimension (G.sideMarginal x S) + 2) ≤ _ at hb1 |
| change 2 * ((G.onSide Sᶜ).faceCodimension (G.sideMarginal x Sᶜ) + 2) ≤ _ at hb2 |
| omega |
| · have hb := G.faceCodimension_no_tight heven' hcut |
| omega |
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| theorem sharp_face_rank (G : LooplessGraph V E) (m : ℕ) |
| (hm : 0 < m) (hV : Fintype.card V = 2 * m) {x : E → ℝ} |
| (hx : x ∈ G.polytope) : G.faceCodimension x + 2 ≤ 3 * m := by |
| by_cases hm1 : m = 1 |
| · have htwo : Fintype.card V = 2 := by omega |
| have hb := faceCodimension_two G htwo x |
| omega |
| · have heven : Even (Fintype.card V) := by rw [hV]; exact even_two_mul m |
| have hb := sharp_face_rank_aux (Fintype.card V) G rfl heven (by omega) x |
| (G.polytope_subset_constraintPolytope hx) |
| omega |
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| end MatchingEntropyBounds.Refined |
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| end |
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| end OAI |
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