| import OAI.Combinatorics.MatchingEntropy.KleeSharpness |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators Topology |
| open Filter |
| |
| universe uV uE uI uJ uA uB uC uW uF uD uι |
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| namespace MatchingEntropyBounds.Refined |
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| def obstructionGraph : LooplessGraph (Fin 8) (Fin 12) where |
| left := ![0,0,0,1,1,2,2,2,3,5,5,6] |
| right := ![1,3,5,4,6,3,4,7,4,6,7,7] |
| loopless := by intro e; fin_cases e <;> decide |
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| def obstructionEdges (i : Fin 5) : Finset (Fin 12) := |
| ![{0,7,8,9}, {1,4,6,10}, {2,3,5,11}, {1,3,7,9}, {2,4,7,8}] i |
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| def obstructionUniform : Fin 12 → ℝ := fun _ => 1 / 3 |
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| def obstructionLaw : obstructionGraph.Matching → ℝ := |
| fun M => ∑ i : Fin 3, if M.val = obstructionEdges i.castSucc.castSucc then 1 / 3 else 0 |
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| theorem obstruction_simple : IsSimpleLabelled obstructionGraph := by |
| unfold IsSimpleLabelled obstructionGraph |
| decide |
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| theorem obstruction_matching (i : Fin 5) : obstructionGraph.IsPerfectMatching (obstructionEdges i) := by |
| unfold LooplessGraph.IsPerfectMatching LooplessGraph.Incident obstructionGraph ExistsUnique |
| fin_cases i <;> decide |
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| def obstructionMatching (i : Fin 5) : obstructionGraph.Matching := ⟨obstructionEdges i,obstruction_matching i⟩ |
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| theorem obstructionEdges_injective : Function.Injective obstructionEdges := by |
| unfold Function.Injective obstructionEdges |
| decide |
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| theorem obstructionMatching_injective : Function.Injective obstructionMatching := by |
| intro i j h |
| exact obstructionEdges_injective (congrArg Subtype.val h) |
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|
| theorem obstruction_binary_classify (a b c d e f g h i j k l : ℕ) |
| (h0 : a+b+c=1) (h1 : a+d+e=1) (h2 : f+g+h=1) (h3 : b+f+i=1) |
| (h4 : d+g+i=1) (h5 : c+j+k=1) (h6 : e+j+l=1) (h7 : h+k+l=1) : |
| (a=1 ∧ b=0 ∧ c=0 ∧ d=0 ∧ e=0 ∧ f=0 ∧ g=0 ∧ h=1 ∧ i=1 ∧ j=1 ∧ k=0 ∧ l=0) ∨ |
| (a=0 ∧ b=1 ∧ c=0 ∧ d=0 ∧ e=1 ∧ f=0 ∧ g=1 ∧ h=0 ∧ i=0 ∧ j=0 ∧ k=1 ∧ l=0) ∨ |
| (a=0 ∧ b=0 ∧ c=1 ∧ d=1 ∧ e=0 ∧ f=1 ∧ g=0 ∧ h=0 ∧ i=0 ∧ j=0 ∧ k=0 ∧ l=1) ∨ |
| (a=0 ∧ b=1 ∧ c=0 ∧ d=1 ∧ e=0 ∧ f=0 ∧ g=0 ∧ h=1 ∧ i=0 ∧ j=1 ∧ k=0 ∧ l=0) ∨ |
| (a=0 ∧ b=0 ∧ c=1 ∧ d=0 ∧ e=1 ∧ f=0 ∧ g=0 ∧ h=1 ∧ i=1 ∧ j=0 ∧ k=0 ∧ l=0) := by |
| rcases (show a = 1 ∨ b = 1 ∨ c = 1 by omega) with ha | hb | hc |
| · left |
| omega |
| · rcases (show d = 0 ∨ d = 1 by omega) with hd | hd |
| · right; left; omega |
| · right; right; right; left; omega |
| · rcases (show d = 0 ∨ d = 1 by omega) with hd | hd |
| · right; right; right; right; omega |
| · right; right; left; omega |
|
|
| theorem obstruction_classify (M : obstructionGraph.Matching) : ∃ i : Fin 5, M = obstructionMatching i := by |
| let a : Fin 12 → ℕ := fun e => if e ∈ M.val then 1 else 0 |
| have hadeg (v : Fin 8) : (∑ e : Fin 12, if obstructionGraph.Incident v e then a e else 0) = 1 := by |
| obtain ⟨e,he,hu⟩ := M.property v |
| rw [Finset.sum_eq_single e] |
| · simp [he,a] |
| · intro f _ hfe |
| by_cases hf : obstructionGraph.Incident v f |
| · have hfm : f ∉ M.val := fun h => hfe (hu f ⟨h,hf⟩) |
| simp [hf,a,hfm] |
| · simp [hf] |
| · simp |
| have h0 := hadeg 0 |
| have h1 := hadeg 1 |
| have h2 := hadeg 2 |
| have h3 := hadeg 3 |
| have h4 := hadeg 4 |
| have h5 := hadeg 5 |
| have h6 := hadeg 6 |
| have h7 := hadeg 7 |
| simp +decide [obstructionGraph,LooplessGraph.Incident,Fin.sum_univ_succ] at h0 h1 h2 h3 h4 h5 h6 h7 |
| have hcases := obstruction_binary_classify (a 0) (a 1) (a 2) (a 3) (a 4) (a 5) |
| (a 6) (a 7) (a 8) (a 9) (a 10) (a 11) (by omega) (by omega) (by omega) (by omega) |
| (by omega) (by omega) (by omega) (by omega) |
| clear hadeg h0 h1 h2 h3 h4 h5 h6 h7 |
| rcases hcases with hh | hh | hh | hh | hh |
| all_goals simp [a] at hh |
| · refine ⟨0,Subtype.ext ?_⟩ |
| ext e |
| fin_cases e <;> simp_all +decide |
| · refine ⟨1,Subtype.ext ?_⟩ |
| ext e |
| fin_cases e <;> simp_all +decide |
| · refine ⟨2,Subtype.ext ?_⟩ |
| ext e |
| fin_cases e <;> simp_all +decide |
| · refine ⟨3,Subtype.ext ?_⟩ |
| ext e |
| fin_cases e <;> simp_all +decide |
| · refine ⟨4,Subtype.ext ?_⟩ |
| ext e |
| fin_cases e <;> simp_all +decide |
|
|
| def obstructionEquiv : Fin 5 ≃ obstructionGraph.Matching := |
| Equiv.ofBijective obstructionMatching ⟨obstructionMatching_injective, |
| fun M => by obtain ⟨i,hi⟩ := obstruction_classify M; exact ⟨i,hi.symm⟩⟩ |
|
|
| def obstructionSelected (i : Fin 3) : obstructionGraph.Matching := |
| obstructionMatching (i.castSucc.castSucc) |
|
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| theorem obstructionSelected_injective : Function.Injective obstructionSelected := by |
| intro i j h |
| exact Fin.ext (congrArg (fun k : Fin 5 => k.val) (obstructionMatching_injective h)) |
|
|
| theorem obstruction_law_eq_fiber : |
| obstructionLaw = fiberMass (uniformLaw (Fin 3)) obstructionSelected := by |
| ext M |
| unfold obstructionLaw fiberMass |
| apply Finset.sum_congr rfl |
| intro i _ |
| have he : obstructionSelected i = M ↔ M.val = obstructionEdges i.castSucc.castSucc := by |
| rw [Subtype.ext_iff] |
| exact eq_comm |
| simp [he,uniformLaw] |
|
|
| theorem obstruction_law_mean : obstructionGraph.mean obstructionLaw = obstructionUniform := by |
| ext e |
| rw [obstruction_law_eq_fiber,LooplessGraph.mean,fiberMass_expect] |
| fin_cases e <;> simp +decide [obstructionSelected,obstructionMatching,obstructionEdges, |
| uniformLaw,LooplessGraph.indicator,obstructionUniform,Fin.sum_univ_succ] |
|
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| theorem obstruction_law_feasible : obstructionLaw ∈ obstructionGraph.feasibleLaws obstructionUniform := by |
| refine ⟨?_,obstruction_law_mean⟩ |
| rw [obstruction_law_eq_fiber] |
| exact fiberMass_mem_simplex uniformLaw_simplex _ |
|
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| theorem obstruction_uniform_mem : obstructionUniform ∈ obstructionGraph.polytope := |
| (obstructionGraph.mem_polytope_iff _).mpr ⟨_,obstruction_law_feasible⟩ |
|
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| theorem obstruction_mean_coordinates (p : obstructionGraph.Matching → ℝ) : |
| obstructionGraph.mean p = |
| ![p (obstructionMatching 0), p (obstructionMatching 1) + p (obstructionMatching 3), |
| p (obstructionMatching 2) + p (obstructionMatching 4), |
| p (obstructionMatching 2) + p (obstructionMatching 3), |
| p (obstructionMatching 1) + p (obstructionMatching 4), |
| p (obstructionMatching 2), p (obstructionMatching 1), |
| p (obstructionMatching 0) + p (obstructionMatching 3) + p (obstructionMatching 4), |
| p (obstructionMatching 0) + p (obstructionMatching 4), |
| p (obstructionMatching 0) + p (obstructionMatching 3), |
| p (obstructionMatching 1), p (obstructionMatching 2)] := by |
| ext e |
| unfold LooplessGraph.mean |
| rw [← Equiv.sum_comp obstructionEquiv] |
| change (∑ i : Fin 5, p (obstructionMatching i) * obstructionGraph.indicator (obstructionMatching i) e) = _ |
| fin_cases e <;> simp +decide [LooplessGraph.indicator,obstructionMatching,obstructionEdges,Fin.sum_univ_succ,add_assoc] |
|
|
| theorem obstruction_law_values (i : Fin 5) : |
| obstructionLaw (obstructionMatching i) = ![(1 : ℝ)/3,1/3,1/3,0,0] i := by |
| fin_cases i <;> simp +decide [obstructionLaw,obstructionMatching,obstructionEdges,Fin.sum_univ_succ] |
|
|
| theorem obstruction_unique_law : |
| obstructionGraph.feasibleLaws obstructionUniform = {obstructionLaw} := by |
| apply Set.Subset.antisymm ?_ (Set.singleton_subset_iff.mpr obstruction_law_feasible) |
| intro p hp |
| have hm := hp.2 |
| rw [obstruction_mean_coordinates] at hm |
| have h0 := congrFun hm 0 |
| have h1 := congrFun hm 6 |
| have h2 := congrFun hm 5 |
| have h3 := congrFun hm 3 |
| have h4 := congrFun hm 4 |
| change p (obstructionMatching 0) = 1/3 at h0 |
| change p (obstructionMatching 1) = 1/3 at h1 |
| change p (obstructionMatching 2) = 1/3 at h2 |
| change p (obstructionMatching 2) + p (obstructionMatching 3) = 1/3 at h3 |
| change p (obstructionMatching 1) + p (obstructionMatching 4) = 1/3 at h4 |
| apply Set.mem_singleton_iff.mpr |
| ext M |
| obtain ⟨i,rfl⟩ := obstruction_classify M |
| rw [obstruction_law_values] |
| fin_cases i |
| · exact h0 |
| · exact h1 |
| · exact h2 |
| · change p (obstructionMatching 3) = 0 |
| linarith only [h2,h3] |
| · change p (obstructionMatching 4) = 0 |
| linarith only [h1,h4] |
|
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| theorem obstruction_entropy_law : entropy obstructionLaw = Real.log 3 := by |
| rw [obstruction_law_eq_fiber,entropy_fiberMass_injective _ _ obstructionSelected_injective,uniformLaw_entropy] |
| norm_num |
|
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| theorem obstruction_maxEntropy : maxMatchingEntropy obstructionGraph obstructionUniform = Real.log 3 := by |
| obtain ⟨p,hp,he⟩ := maxMatchingEntropy_attained _ obstruction_uniform_mem |
| rw [obstruction_unique_law,Set.mem_singleton_iff] at hp |
| rw [← he,hp,obstruction_entropy_law] |
|
|
| theorem obstruction_entropy_terms : |
| entropy obstructionUniform = 4 * Real.log 3 ∧ |
| complementEntropy obstructionUniform = 8 * Real.log (3/2) := by |
| have hF : Real.negMulLog (1/3) = Real.log 3 / 3 := by |
| norm_num [Real.negMulLog,Real.log_div] |
| ring |
| have hB : Real.negMulLog (1-1/3) = (2/3) * Real.log (3/2) := by |
| norm_num [Real.negMulLog,Real.log_div] |
| ring |
| constructor |
| · simp only [entropy,obstructionUniform,hF,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
| ring |
| · change (∑ _ : Fin 12, Real.negMulLog (1-1/3)) = _ |
| simp only [hB,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
| ring |
|
|
| theorem obstruction_logarithms : |
| entropy obstructionUniform - complementEntropy obstructionUniform = 4 * Real.log (4/3) ∧ |
| entropy obstructionUniform - complementEntropy obstructionUniform - |
| maxMatchingEntropy obstructionGraph obstructionUniform = Real.log (256/243) := by |
| have h4 : Real.log 4 = 2 * Real.log 2 := by |
| convert Real.log_pow (2 : ℝ) 2 using 1 |
| norm_num |
| have h256 : Real.log 256 = 8 * Real.log 2 := by |
| convert Real.log_pow (2 : ℝ) 8 using 1 |
| norm_num |
| have h243 : Real.log 243 = 5 * Real.log 3 := by |
| convert Real.log_pow (3 : ℝ) 5 using 1 |
| norm_num |
| rw [obstruction_entropy_terms.1,obstruction_entropy_terms.2,obstruction_maxEntropy] |
| norm_num [Real.log_div,h4,h256,h243] |
| constructor <;> ring |
|
|
| theorem eight_vertex_obstruction : |
| IsSimpleLabelled obstructionGraph ∧ |
| (∀ M : obstructionGraph.Matching, ∃! i : Fin 5, M.val = obstructionEdges i) ∧ |
| (∀ i : Fin 5, obstructionGraph.IsPerfectMatching (obstructionEdges i)) ∧ |
| Fintype.card obstructionGraph.Matching = 5 ∧ |
| obstructionUniform ∈ obstructionGraph.polytope ∧ |
| obstructionGraph.feasibleLaws obstructionUniform = {obstructionLaw} ∧ |
| maxMatchingEntropy obstructionGraph obstructionUniform = Real.log 3 ∧ |
| entropy obstructionUniform - complementEntropy obstructionUniform = 4 * Real.log (4 / 3) ∧ |
| entropy obstructionUniform - complementEntropy obstructionUniform - |
| maxMatchingEntropy obstructionGraph obstructionUniform = Real.log (256 / 243) ∧ |
| 0 < Real.log (256 / 243) := by |
| refine ⟨obstruction_simple,?_,obstruction_matching,?_,obstruction_uniform_mem, |
| obstruction_unique_law,obstruction_maxEntropy,obstruction_logarithms.1, |
| obstruction_logarithms.2,Real.log_pos (by norm_num)⟩ |
| · intro M |
| obtain ⟨i,rfl⟩ := obstruction_classify M |
| refine ⟨i,rfl,?_⟩ |
| intro j hj |
| exact (obstructionEdges_injective hj).symm |
| · exact (Fintype.card_congr obstructionEquiv).symm.trans (Fintype.card_fin 5) |
|
|
| end MatchingEntropyBounds.Refined |
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| end |
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| end OAI |
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