| import OAI.Combinatorics.PerfectMatching.CompletePairs |
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| namespace OAI |
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| section |
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| namespace BinaryMatching |
| noncomputable section |
| open Finset MatchingEntropy |
| variable {n : ℕ} |
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| lemma degree_add (x z : Pair n → ℝ) (v : Fin n) : |
| (completeGraph n).degree (fun e => x e+z e) v= |
| (completeGraph n).degree x v+(completeGraph n).degree z v := by |
| simpa only [one_smul,one_mul,Pi.add_def] using (completeGraph n).degree_linear x z 1 1 v |
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| lemma cut_add (x z : Pair n → ℝ) (S : Finset (Fin n)) : |
| (completeGraph n).cutMass (fun e => x e+z e) S= |
| (completeGraph n).cutMass x S+(completeGraph n).cutMass z S := by |
| simpa only [one_smul,one_mul,Pi.add_def] using (completeGraph n).cutMass_linear x z 1 1 S |
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| lemma cut_bound {z : Pair n → ℝ} {b : ℝ} (hb : 0≤b) (hz : ∀ e, |z e|≤b) |
| (S : Finset (Fin n)) : |(completeGraph n).cutMass z S|≤Fintype.card (Pair n)*b := by |
| calc |
| _ ≤ ∑ e∈(completeGraph n).cut S, |z e| := abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ _e∈(completeGraph n).cut S, b := sum_le_sum (fun e _ => hz e) |
| _ = ((completeGraph n).cut S).card*b := by simp |
| _ ≤ Fintype.card (Pair n)*b := by |
| exact mul_le_mul_of_nonneg_right (by exact_mod_cast card_le_univ _) hb |
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| def degreeCorrection (x : Pair n → ℝ) (e : Pair n) : ℝ := |
| let d := fun v => 1-(completeGraph n).degree x v |
| (d e.val.1+d e.val.2)/((n:ℝ)-2)-(∑ v,d v)/(((n:ℝ)-2)*((n:ℝ)-1)) |
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| lemma correction_degree (hn : 4≤n) (x : Pair n → ℝ) (v : Fin n) : |
| (completeGraph n).degree (fun e => x e+degreeCorrection x e) v=1 := by |
| have hn' : (4:ℝ)≤n := by exact_mod_cast hn |
| have h2 : (n:ℝ)-2≠0 := by linarith |
| have h1 : (n:ℝ)-1≠0 := by linarith |
| let d := fun v => 1-(completeGraph n).degree x v |
| rw [degree_add] |
| have hc : degreeCorrection x = ((n:ℝ)-2)⁻¹ • (fun e : Pair n => d e.val.1+d e.val.2) + |
| (-1:ℝ) • (fun _ => (∑ v,d v)/(((n:ℝ)-2)*((n:ℝ)-1))) := by |
| funext e; simp only [degreeCorrection,d,Pi.add_apply,Pi.smul_apply,smul_eq_mul]; ring |
| rw [hc,(completeGraph n).degree_linear,degree_vertex_add,degree_constant] |
| dsimp [d] |
| field_simp |
| ring |
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| lemma correction_bound (hn : 4≤n) {x : Pair n → ℝ} {w : ℝ} (hw : 0≤w) |
| (hdeg : ∀ v, |(completeGraph n).degree x v-1|≤w) (e : Pair n) : |
| |degreeCorrection x e|≤2*w := by |
| have hn' : (4:ℝ)≤n := by exact_mod_cast hn |
| have h2 : 0<(n:ℝ)-2 := by linarith |
| have h1 : 0<(n:ℝ)-1 := by linarith |
| let d := fun v => 1-(completeGraph n).degree x v |
| have hd (v) : |d v|≤w := by dsimp [d]; rw [abs_sub_comm]; exact hdeg v |
| have hsum : |∑ v,d v|≤(n:ℝ)*w := by |
| calc |
| _ ≤ ∑ v,|d v| := abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ _v : Fin n,w := sum_le_sum (fun v _ => hd v) |
| _ = _ := by simp |
| have ha : |(d e.val.1+d e.val.2)/((n:ℝ)-2)|≤w := by |
| rw [abs_div,abs_of_pos h2,div_le_iff₀ h2] |
| have hh := abs_add_le (d e.val.1) (d e.val.2) |
| nlinarith [hd e.val.1,hd e.val.2] |
| have hb : |(∑ v,d v)/(((n:ℝ)-2)*((n:ℝ)-1))|≤w := by |
| rw [abs_div,abs_of_pos (mul_pos h2 h1),div_le_iff₀ (mul_pos h2 h1)] |
| have hp : (n:ℝ)≤((n:ℝ)-2)*((n:ℝ)-1) := by nlinarith |
| exact hsum.trans (by nlinarith [mul_le_mul_of_nonneg_right hp hw]) |
| exact (abs_sub _ _).trans (by dsimp [degreeCorrection] at *; linarith [ha,hb]) |
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| def uniform (n : ℕ) : Pair n → ℝ := fun _ => 1/((n:ℝ)-1) |
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| lemma uniform_box (hn : 4≤n) (e : Pair n) : uniform n e∈Set.Icc (0:ℝ) 1 := by |
| have hn' : (4:ℝ)≤n := by exact_mod_cast hn |
| constructor |
| · exact div_nonneg (by norm_num) (by linarith) |
| · exact (div_le_one (by linarith)).2 (by linarith) |
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| lemma uniform_degree (hn : 4≤n) (v : Fin n) : |
| (completeGraph n).degree (uniform n) v=1 := by |
| unfold uniform |
| rw [degree_constant] |
| have hn' : (4:ℝ)≤n := by exact_mod_cast hn |
| field_simp [show (n:ℝ)-1≠0 by linarith] |
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| def TrivialCut (S : Finset (Fin n)) : Prop := S.card=1 ∨ Sᶜ.card=1 |
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| lemma trivial_cut_degree {x : Pair n → ℝ} (hx : ∀ v,(completeGraph n).degree x v=1) |
| {S : Finset (Fin n)} (hS : TrivialCut S) : (completeGraph n).cutMass x S=1 := by |
| rcases hS with h|h |
| · obtain ⟨v,rfl⟩ := card_eq_one.mp h |
| simpa using hx v |
| · obtain ⟨v,hv⟩ := card_eq_one.mp h |
| have he : (completeGraph n).cutMass x S=(completeGraph n).cutMass x Sᶜ := by |
| simp only [LooplessGraph.cutMass,LooplessGraph.cut_compl] |
| rw [he,hv,LooplessGraph.cutMass_singleton,hx] |
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| lemma uniform_nontrivial_cut (heven : Even n) {S : Finset (Fin n)} (hodd : Odd S.card) |
| (hS : ¬TrivialCut S) : (3/2:ℝ)≤(completeGraph n).cutMass (uniform n) S := by |
| have hcards : S.card+Sᶜ.card=n := by simp |
| have ha : 3≤S.card := by |
| have ho := Nat.odd_iff.mp hodd |
| have hh : S.card≠1 := fun h => hS (Or.inl h) |
| omega |
| have hb : 3≤Sᶜ.card := by |
| have ho := Nat.odd_iff.mp hodd |
| have he := Nat.even_iff.mp heven |
| have hh : Sᶜ.card≠1 := fun h => hS (Or.inr h) |
| omega |
| have ha' : (3:ℝ)≤S.card := by exact_mod_cast ha |
| have hb' : (3:ℝ)≤Sᶜ.card := by exact_mod_cast hb |
| have hc' : (S.card:ℝ)+Sᶜ.card=n := by exact_mod_cast hcards |
| have hn : (6:ℝ)≤n := by linarith |
| unfold uniform |
| rw [cut_constant] |
| rw [mul_one_div,le_div_iff₀ (by linarith : 0<(n:ℝ)-1)] |
| have hp := mul_nonneg (sub_nonneg.mpr ha') (sub_nonneg.mpr hb') |
| nlinarith |
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| lemma uniform_feasible (hn : 4≤n) (heven : Even n) : |
| uniform n∈(completeGraph n).polytope := by |
| rw [(completeGraph n).polytope_eq_constraints (by simpa using heven)] |
| refine ⟨fun e => (uniform_box hn e).1,uniform_degree hn,?_⟩ |
| intro S hS |
| by_cases ht : TrivialCut S |
| · rw [trivial_cut_degree (uniform_degree hn) ht] |
| · linarith [uniform_nontrivial_cut heven hS ht] |
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| theorem feasibility_repair (hn : 4≤n) (heven : Even n) {x : Pair n → ℝ} |
| (hx : ∀ e,x e∈Set.Icc (0:ℝ) 1) {w : ℝ} (hw : 0≤w) |
| (hdeg : ∀ v, |(completeGraph n).degree x v-1|≤w) |
| (hcut : ∀ S : Finset (Fin n),Odd S.card → 1-w≤(completeGraph n).cutMass x S) : |
| ∃ y∈(completeGraph n).polytope,∀ e,|y e-x e|≤(2+10*(n:ℝ)^2)*w := by |
| let A : ℝ := 10*(n:ℝ)^2 |
| let c := degreeCorrection x |
| let x' := fun e => x e+c e |
| let t := min 1 (A*w) |
| let y : Pair n → ℝ := (1-t) • x'+t • uniform n |
| have hn' : (4:ℝ)≤n := by exact_mod_cast hn |
| have hA : 0≤A := by dsimp [A]; positivity |
| have ht : 0≤t := le_min (by norm_num) (mul_nonneg hA hw) |
| have ht1 : t≤1 := min_le_left _ _ |
| have htw : t≤A*w := min_le_right _ _ |
| have hc := correction_bound hn hw hdeg |
| have hx'd := correction_degree hn x |
| have hx'lo (e) : -2*w≤x' e := by |
| have hh := (abs_le.mp (hc e)).1 |
| dsimp [x',c]; linarith [(hx e).1] |
| have hx'cut (S) (hS : Odd S.card) : |
| 1-(1+2*(Fintype.card (Pair n):ℝ))*w≤(completeGraph n).cutMass x' S := by |
| have hh := (abs_le.mp (cut_bound (mul_nonneg (by norm_num) hw) hc S)).1 |
| dsimp [x',c]; rw [cut_add] |
| linarith [hcut S hS] |
| have hA1 : 2≤A/((n:ℝ)-1) := by |
| apply (le_div_iff₀ (by linarith)).2 |
| dsimp [A]; nlinarith |
| have hA2 : 1+2*(Fintype.card (Pair n):ℝ)≤A/2 := by |
| have hq : (Fintype.card (Pair n):ℝ)≤(n:ℝ)^2 := by exact_mod_cast q_le_square (n:=n) |
| dsimp [A]; nlinarith |
| have hyd : ∀ v,(completeGraph n).degree y v=1 := by |
| intro v; dsimp [y]; rw [LooplessGraph.degree_linear,hx'd,uniform_degree hn]; ring |
| have hy : y∈(completeGraph n).polytope := by |
| by_cases hbig : 1≤A*w |
| · have htt : t=1 := min_eq_left hbig |
| simpa only [y,htt,sub_self,zero_smul,one_smul,zero_add] using uniform_feasible hn heven |
| · have htt : t=A*w := min_eq_right (le_of_not_ge hbig) |
| rw [(completeGraph n).polytope_eq_constraints (by simpa using heven)] |
| refine ⟨?_,hyd,?_⟩ |
| · intro e |
| have h0 := mul_le_mul_of_nonneg_left (hx'lo e) (sub_nonneg.mpr ht1) |
| have h1 := mul_le_mul_of_nonneg_right hA1 hw |
| change 0≤(1-t)*x' e+t*uniform n e |
| unfold uniform |
| rw [htt] at * |
| simp only [div_eq_mul_inv] at h1 ⊢ |
| nlinarith [mul_nonneg ht hw] |
| · intro S hS |
| by_cases htr : TrivialCut S |
| · rw [trivial_cut_degree hyd htr] |
| · have h0 := mul_le_mul_of_nonneg_left (hx'cut S hS) (sub_nonneg.mpr ht1) |
| have h1 := mul_le_mul_of_nonneg_left (uniform_nontrivial_cut heven hS htr) ht |
| have h2 := mul_le_mul_of_nonneg_right hA2 hw |
| dsimp [y]; rw [LooplessGraph.cutMass_linear] |
| have hqw : 0≤(1+2*(Fintype.card (Pair n):ℝ))*w := by positivity |
| rw [htt] at * |
| nlinarith [mul_nonneg ht hqw] |
| refine ⟨y,hy,fun e => ?_⟩ |
| have hu : |uniform n e-x e|≤1 := by |
| rw [abs_le]; constructor <;> linarith [(uniform_box hn e).1,(uniform_box hn e).2,(hx e).1,(hx e).2] |
| have he : y e-x e=(1-t)*c e+t*(uniform n e-x e) := by dsimp [y,x']; ring |
| rw [he] |
| calc |
| _ ≤ |(1-t)*c e|+|t*(uniform n e-x e)| := abs_add_le _ _ |
| _ = (1-t)*|c e|+t*|uniform n e-x e| := by rw [abs_mul,abs_mul,abs_of_nonneg (sub_nonneg.mpr ht1),abs_of_nonneg ht] |
| _ ≤ (1-t)*(2*w)+t*1 := add_le_add (mul_le_mul_of_nonneg_left (hc e) (sub_nonneg.mpr ht1)) (mul_le_mul_of_nonneg_left hu ht) |
| _ ≤ (2+10*(n:ℝ)^2)*w := by dsimp [A] at htw; nlinarith [mul_nonneg ht hw] |
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| end |
| end BinaryMatching |
| end |
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| end OAI |
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