| import Mathlib | |
| namespace OAI | |
| namespace MatchingFPRAS.Cells | |
| open SimpleGraph | |
| variable {L : Type*} {G : SimpleGraph L} | |
| lemma exists_walk_on_trace (a : ℕ → L) (n : ℕ) | |
| (ha : ∀ i < n, a i = a (i + 1) ∨ G.Adj (a i) (a (i + 1))) : | |
| ∃ w : G.Walk (a 0) (a n), ∀ v ∈ w.support, ∃ i ≤ n, a i = v := by | |
| induction n with | |
| | zero => | |
| refine ⟨.nil, ?_⟩ | |
| intro v hv | |
| have hv' : v = a 0 := by simpa using hv | |
| exact ⟨0, le_rfl, hv'.symm⟩ | |
| | succ n ih => | |
| obtain ⟨w, hw⟩ := ih (fun i hi => ha i (by omega)) | |
| rcases ha n (by omega) with heq | hadj | |
| · refine ⟨w.copy rfl heq, ?_⟩ | |
| intro v hv | |
| obtain ⟨i, hi, hiv⟩ := hw v (by simpa using hv) | |
| exact ⟨i, by omega, hiv⟩ | |
| · refine ⟨w.concat hadj, ?_⟩ | |
| intro v hv | |
| have hv' : v ∈ w.support ∨ v = a (n + 1) := by simpa using hv | |
| rcases hv' with hv' | hv' | |
| · obtain ⟨i, hi, hiv⟩ := hw v hv' | |
| exact ⟨i, by omega, hiv⟩ | |
| · exact ⟨n + 1, le_rfl, hv'.symm⟩ | |
| lemma neighbor_return (hG : G.IsAcyclic) (a : ℕ → L) (n : ℕ) (P : L) | |
| (ha : ∀ i < n, a i = a (i + 1) ∨ G.Adj (a i) (a (i + 1))) | |
| (havoid : ∀ i ≤ n, a i ≠ P) (hfirst : G.Adj P (a 0)) | |
| (hlast : G.Adj P (a n)) : a 0 = a n := by | |
| classical | |
| obtain ⟨w, hw⟩ := exists_walk_on_trace a n ha | |
| have hnot : P ∉ w.toPath.val.support := by | |
| intro hP | |
| obtain ⟨i, hi, hiP⟩ := hw P (w.support_toPath_subset_support hP) | |
| exact havoid i hi hiP | |
| have hp : (Walk.cons hfirst w.toPath.val).IsPath := w.toPath.isPath.cons hnot | |
| have hv := hG.eq_snd_of_adj_start hp hlast (Walk.cons hfirst w.toPath.val).end_mem_support | |
| simpa using hv.symm | |
| structure ArcData (G : SimpleGraph L) (s : ℕ) where | |
| labels : ℕ → L | |
| sets : ℕ → Finset L | |
| left_mem : ∀ i < s, labels i ∈ sets i | |
| right_mem : ∀ i < s, labels i ∈ sets (i + 1) | |
| clique : ∀ i ≤ s, ∀ u ∈ sets i, ∀ v ∈ sets i, u = v ∨ G.Adj u v | |
| namespace ArcData | |
| variable {s : ℕ} (A : ArcData G s) | |
| def Admissible (x y : ℕ) : Prop := ∃ l, l ∈ A.sets x ∧ l ∈ A.sets y | |
| def Good (x y : ℕ) : Prop := y = x + 1 ∨ A.Admissible (x + 1) (y - 1) | |
| structure Split (exteriorGood : Prop) where | |
| p₁ : ℕ | |
| p₂ : ℕ | |
| positive : 0 < p₁ | |
| ordered : p₁ < p₂ | |
| inside : p₂ < s | |
| odd₁ : p₁ % 2 = 1 | |
| even₂ : p₂ % 2 = 0 | |
| first : A.Admissible 0 p₁ | |
| middle : A.Admissible p₁ p₂ | |
| third : A.Admissible p₂ s | |
| some_good : (A.Good 0 p₁ ∧ A.Good p₂ s) ∨ (A.Good p₁ p₂ ∧ exteriorGood) | |
| outer_long : 1 < p₁ → 1 < s - p₂ → A.Good p₁ p₂ ∧ exteriorGood | |
| middle_long : 1 < p₂ - p₁ → A.Good 0 p₁ ∧ A.Good p₂ s | |
| lemma trace (i : ℕ) (hi : i + 1 < s) : | |
| A.labels i = A.labels (i + 1) ∨ G.Adj (A.labels i) (A.labels (i + 1)) := | |
| A.clique (i + 1) (by omega) _ (A.right_mem i (by omega)) _ (A.left_mem (i + 1) hi) | |
| lemma edge_admissible (i : ℕ) (hi : i < s) : A.Admissible i (i + 1) := | |
| ⟨A.labels i, A.left_mem i hi, A.right_mem i hi⟩ | |
| lemma last_mem (hs : 0 < s) : A.labels (s - 1) ∈ A.sets s := by | |
| convert A.right_mem (s - 1) (by omega) using 1 | |
| congr 1 | |
| omega | |
| lemma good_of_same_boundary (x y : ℕ) (hxy : x < y) (hy : y ≤ s) | |
| (heq : A.labels x = A.labels (y - 1)) : A.Good x y := by | |
| right | |
| refine ⟨A.labels x, A.right_mem x (by omega), ?_⟩ | |
| rw [heq] | |
| exact A.left_mem (y - 1) (by omega) | |
| lemma neighbor_return_between (hG : G.IsAcyclic) (P : L) | |
| (lo hi : ℕ) (hlo : lo ≤ hi) (hhi : hi < s) | |
| (havoid : ∀ i, lo ≤ i → i ≤ hi → A.labels i ≠ P) | |
| (hfirst : G.Adj P (A.labels lo)) (hlast : G.Adj P (A.labels hi)) : | |
| A.labels lo = A.labels hi := by | |
| have hf : G.Adj P ((fun i => A.labels (lo + i)) 0) := by simpa using hfirst | |
| have hl : G.Adj P ((fun i => A.labels (lo + i)) (hi - lo)) := by | |
| simpa only [Nat.add_sub_of_le hlo] using hlast | |
| have h := neighbor_return hG (fun i => A.labels (lo + i)) (hi - lo) P | |
| (fun i hi' => by | |
| have := A.trace (lo + i) (by omega) | |
| simpa only [Nat.add_assoc] using this) | |
| (fun i hi' => havoid (lo + i) (by omega) (by omega)) hf hl | |
| simpa only [Nat.add_zero, Nat.add_sub_of_le hlo] using h | |
| lemma split_short_middle {exteriorGood : Prop} (hext : exteriorGood) | |
| (p : ℕ) (hp : 0 < p) (hps : p + 1 < s) (hodd : p % 2 = 1) | |
| (hfirst : A.Admissible 0 p) (hthird : A.Admissible (p + 1) s) : | |
| Nonempty (A.Split exteriorGood) := by | |
| refine ⟨⟨p, p + 1, hp, by omega, hps, hodd, by omega, | |
| hfirst, A.edge_admissible p (by omega), hthird, ?_, ?_, ?_⟩⟩ | |
| · exact Or.inr ⟨Or.inl rfl, hext⟩ | |
| · exact fun _ _ => ⟨Or.inl rfl, hext⟩ | |
| · intro h | |
| omega | |
| lemma split_short_outer (exteriorGood : Prop) (hs : 3 ≤ s) (hsodd : s % 2 = 1) | |
| (heq : A.labels 0 = A.labels (s - 1)) : Nonempty (A.Split exteriorGood) := by | |
| have hg₁ : A.Good 0 1 := Or.inl rfl | |
| have hg₃ : A.Good (s - 1) s := Or.inl (by omega) | |
| have ha₃ : A.Admissible (s - 1) s := | |
| ⟨A.labels (s - 1), A.left_mem (s - 1) (by omega), A.last_mem (by omega)⟩ | |
| have ha₂ : A.Admissible 1 (s - 1) := by | |
| refine ⟨A.labels 0, A.right_mem 0 (by omega), ?_⟩ | |
| rw [heq] | |
| exact A.left_mem (s - 1) (by omega) | |
| refine ⟨⟨1, s - 1, by omega, by omega, by omega, by omega, by omega, | |
| A.edge_admissible 0 (by omega), ha₂, ha₃, Or.inl ⟨hg₁, hg₃⟩, | |
| ?_, fun _ => ⟨hg₁, hg₃⟩⟩⟩ | |
| intro h | |
| omega | |
| lemma split_good_exterior (hG : G.IsAcyclic) {exteriorGood : Prop} (hext : exteriorGood) | |
| (hs : 3 ≤ s) (hsodd : s % 2 = 1) (P : L) | |
| (hP₀ : P ∈ A.sets 0) (hPₛ : P ∈ A.sets s) : Nonempty (A.Split exteriorGood) := by | |
| classical | |
| by_cases hboundary : A.labels 0 = A.labels (s - 1) | |
| · exact A.split_short_outer exteriorGood hs hsodd hboundary | |
| by_cases hodd : ∃ i, i < s ∧ i % 2 = 1 ∧ A.labels i = P | |
| · obtain ⟨i, his, hi, hiP⟩ := hodd | |
| refine A.split_short_middle hext i (by omega) (by omega) hi ?_ ?_ | |
| · exact ⟨P, hP₀, by simpa only [hiP] using A.left_mem i his⟩ | |
| · exact ⟨P, by simpa only [hiP] using A.right_mem i his, hPₛ⟩ | |
| have heven : ∀ i < s, A.labels i = P → i % 2 = 0 := by | |
| intro i his hiP | |
| have hm := Nat.mod_lt i (by omega : 0 < 2) | |
| by_contra hn | |
| exact hodd ⟨i, his, by omega, hiP⟩ | |
| have hex : ∃ i, i < s ∧ A.labels i = P := by | |
| by_contra hnone | |
| push Not at hnone | |
| have hav : ∀ i, 0 ≤ i → i ≤ s - 1 → A.labels i ≠ P := | |
| fun i _ hi => hnone i (by omega) | |
| have hadj₀ : G.Adj P (A.labels 0) := | |
| (A.clique 0 (by omega) P hP₀ _ (A.left_mem 0 (by omega))).resolve_left | |
| (fun h => hnone 0 (by omega) h.symm) | |
| have hadjₛ : G.Adj P (A.labels (s - 1)) := | |
| (A.clique s le_rfl P hPₛ _ (A.last_mem (by omega))).resolve_left | |
| (fun h => hnone (s - 1) (by omega) h.symm) | |
| exact hboundary (A.neighbor_return_between hG P 0 (s - 1) (by omega) (by omega) | |
| hav hadj₀ hadjₛ) | |
| by_cases hzero : A.labels 0 = P | |
| · let j := Nat.findGreatest (fun i => A.labels i = P) (s - 1) | |
| have hjle : j ≤ s - 1 := Nat.findGreatest_le _ | |
| have hjP : A.labels j = P := Nat.findGreatest_spec (P := fun i => A.labels i = P) (by omega : 0 ≤ s - 1) hzero | |
| have hlast : A.labels (s - 1) ≠ P := fun h => hboundary (hzero.trans h.symm) | |
| have hjlt : j < s - 1 := by | |
| by_contra h | |
| have : j = s - 1 := by omega | |
| exact hlast (this ▸ hjP) | |
| have hjEven : j % 2 = 0 := heven j (by omega) hjP | |
| have hav : ∀ i, j + 1 ≤ i → i ≤ s - 1 → A.labels i ≠ P := by | |
| intro i hi his | |
| exact Nat.findGreatest_is_greatest (P := fun k => A.labels k = P) (by omega : j < i) his | |
| have hPa : P ∈ A.sets (j + 1) := by | |
| simpa only [hjP] using A.right_mem j (by omega) | |
| have hadjFirst : G.Adj P (A.labels (j + 1)) := | |
| (A.clique (j + 1) (by omega) P hPa _ (A.left_mem (j + 1) (by omega))).resolve_left | |
| (fun h => hav (j + 1) le_rfl (by omega) h.symm) | |
| have hadjLast : G.Adj P (A.labels (s - 1)) := | |
| (A.clique s le_rfl P hPₛ _ (A.last_mem (by omega))).resolve_left | |
| (fun h => hlast h.symm) | |
| have hQ := A.neighbor_return_between hG P (j + 1) (s - 1) | |
| (by omega) (by omega) hav hadjFirst hadjLast | |
| refine A.split_short_middle hext (j + 1) (by omega) (by omega) (by omega) | |
| ⟨P, hP₀, hPa⟩ ?_ | |
| refine ⟨A.labels (j + 1), A.right_mem (j + 1) (by omega), ?_⟩ | |
| rw [hQ] | |
| exact A.last_mem (by omega) | |
| · let i := Nat.find hex | |
| have hi : i < s := (Nat.find_spec hex).1 | |
| have hiP : A.labels i = P := (Nat.find_spec hex).2 | |
| have hiPos : 0 < i := by | |
| by_contra h | |
| have hi0 : i = 0 := by omega | |
| exact hzero (hi0 ▸ hiP) | |
| have hiEven : i % 2 = 0 := heven i hi hiP | |
| have hia : i - 1 + 1 = i := by omega | |
| have hav : ∀ k, 0 ≤ k → k ≤ i - 1 → A.labels k ≠ P := by | |
| intro k _ hk hkP | |
| have hkmin := Nat.find_min' hex (show k < s ∧ A.labels k = P from ⟨by omega, hkP⟩) | |
| change i ≤ k at hkmin | |
| omega | |
| have hPi : P ∈ A.sets i := by simpa only [hiP] using A.left_mem i hi | |
| have hprev : A.labels (i - 1) ∈ A.sets i := by | |
| simpa only [hia] using A.right_mem (i - 1) (by omega) | |
| have hadjFirst : G.Adj P (A.labels 0) := | |
| (A.clique 0 (by omega) P hP₀ _ (A.left_mem 0 (by omega))).resolve_left | |
| (fun h => hzero h.symm) | |
| have hadjLast : G.Adj P (A.labels (i - 1)) := | |
| (A.clique i (by omega) P hPi _ hprev).resolve_left | |
| (fun h => hav (i - 1) (by omega) le_rfl h.symm) | |
| have hQ := A.neighbor_return_between hG P 0 (i - 1) (by omega) (by omega) | |
| hav hadjFirst hadjLast | |
| refine A.split_short_middle hext (i - 1) (by omega) (by omega) (by omega) ?_ ?_ | |
| · refine ⟨A.labels 0, A.left_mem 0 (by omega), ?_⟩ | |
| rw [hQ] | |
| exact A.left_mem (i - 1) (by omega) | |
| · rw [hia] | |
| exact ⟨P, hPi, hPₛ⟩ | |
| lemma split_left_boundary (hG : G.IsAcyclic) (exteriorGood : Prop) | |
| (hs : 3 ≤ s) (hsodd : s % 2 = 1) (P : L) | |
| (hP₀ : P ∈ A.sets 0) (hPₛ : P ∈ A.sets s) (hzero : A.labels 0 = P) : | |
| Nonempty (A.Split exteriorGood) := by | |
| classical | |
| by_cases hlast : A.labels (s - 1) = P | |
| · exact A.split_short_outer exteriorGood hs hsodd (hzero.trans hlast.symm) | |
| let j := Nat.findGreatest (fun i => A.labels i = P) (s - 1) | |
| have hjle : j ≤ s - 1 := Nat.findGreatest_le _ | |
| have hjP : A.labels j = P := | |
| Nat.findGreatest_spec (P := fun i => A.labels i = P) (by omega : 0 ≤ s - 1) hzero | |
| have hjlt : j < s - 1 := by | |
| by_contra h | |
| have : j = s - 1 := by omega | |
| exact hlast (this ▸ hjP) | |
| have hav : ∀ i, j + 1 ≤ i → i ≤ s - 1 → A.labels i ≠ P := by | |
| intro i hi his | |
| exact Nat.findGreatest_is_greatest (P := fun k => A.labels k = P) (by omega : j < i) his | |
| have hPt : P ∈ A.sets (j + 1) := by | |
| simpa only [hjP] using A.right_mem j (by omega) | |
| have hadjFirst : G.Adj P (A.labels (j + 1)) := | |
| (A.clique (j + 1) (by omega) P hPt _ (A.left_mem (j + 1) (by omega))).resolve_left | |
| (fun h => hav (j + 1) le_rfl (by omega) h.symm) | |
| have hadjLast : G.Adj P (A.labels (s - 1)) := | |
| (A.clique s le_rfl P hPₛ _ (A.last_mem (by omega))).resolve_left | |
| (fun h => hlast h.symm) | |
| have hQ := A.neighbor_return_between hG P (j + 1) (s - 1) | |
| (by omega) (by omega) hav hadjFirst hadjLast | |
| by_cases htEven : (j + 1) % 2 = 0 | |
| · have hg₁ : A.Good 0 1 := Or.inl rfl | |
| have hg₃ : A.Good (j + 1) s := A.good_of_same_boundary _ _ (by omega) le_rfl hQ | |
| have ha₂ : A.Admissible 1 (j + 1) := | |
| ⟨P, by simpa only [hzero] using A.right_mem 0 (by omega), hPt⟩ | |
| have ha₃ : A.Admissible (j + 1) s := by | |
| refine ⟨A.labels (j + 1), A.left_mem (j + 1) (by omega), ?_⟩ | |
| rw [hQ] | |
| exact A.last_mem (by omega) | |
| refine ⟨⟨1, j + 1, by omega, by omega, by omega, by omega, htEven, | |
| A.edge_admissible 0 (by omega), ha₂, ha₃, Or.inl ⟨hg₁, hg₃⟩, | |
| ?_, fun _ => ⟨hg₁, hg₃⟩⟩⟩ | |
| intro h | |
| omega | |
| · have htOdd : (j + 1) % 2 = 1 := by omega | |
| have hg₁ : A.Good 0 (j + 1) := by | |
| apply A.good_of_same_boundary _ _ (by omega) (by omega) | |
| simpa only [Nat.add_sub_cancel, hzero] using hjP.symm | |
| have hg₃ : A.Good (s - 1) s := Or.inl (by omega) | |
| have ha₁ : A.Admissible 0 (j + 1) := ⟨P, hP₀, hPt⟩ | |
| have ha₂ : A.Admissible (j + 1) (s - 1) := by | |
| refine ⟨A.labels (j + 1), A.left_mem (j + 1) (by omega), ?_⟩ | |
| rw [hQ] | |
| exact A.left_mem (s - 1) (by omega) | |
| have ha₃ : A.Admissible (s - 1) s := | |
| ⟨A.labels (s - 1), A.left_mem (s - 1) (by omega), A.last_mem (by omega)⟩ | |
| refine ⟨⟨j + 1, s - 1, by omega, by omega, by omega, htOdd, by omega, | |
| ha₁, ha₂, ha₃, Or.inl ⟨hg₁, hg₃⟩, ?_, fun _ => ⟨hg₁, hg₃⟩⟩⟩ | |
| intro _ h | |
| omega | |
| def reverse : ArcData G s where | |
| labels i := A.labels (s - 1 - i) | |
| sets i := A.sets (s - i) | |
| left_mem i hi := by | |
| have hj : s - 1 - i < s := by omega | |
| have heq : s - 1 - i + 1 = s - i := by omega | |
| simpa only [heq] using A.right_mem (s - 1 - i) hj | |
| right_mem i hi := by | |
| have hj : s - 1 - i < s := by omega | |
| have heq : s - 1 - i = s - (i + 1) := by omega | |
| simpa only [← heq] using A.left_mem (s - 1 - i) hj | |
| clique i hi := A.clique (s - i) (by omega) | |
| lemma admissible_symm {x y : ℕ} (h : A.Admissible x y) : A.Admissible y x := by | |
| obtain ⟨l, hl₁, hl₂⟩ := h | |
| exact ⟨l, hl₂, hl₁⟩ | |
| lemma good_reverse {x y : ℕ} (hxy : x < y) (hy : y ≤ s) | |
| (h : A.reverse.Good x y) : A.Good (s - y) (s - x) := by | |
| rcases h with hshort | ⟨l, hl₁, hl₂⟩ | |
| · left | |
| omega | |
| · right | |
| change l ∈ A.sets (s - (x + 1)) at hl₁ | |
| change l ∈ A.sets (s - (y - 1)) at hl₂ | |
| have h₁ : s - (x + 1) = s - x - 1 := by omega | |
| have h₂ : s - (y - 1) = s - y + 1 := by omega | |
| exact ⟨l, by simpa only [h₂] using hl₂, by simpa only [h₁] using hl₁⟩ | |
| def Split.unreverse {exteriorGood : Prop} (hsodd : s % 2 = 1) | |
| (T : A.reverse.Split exteriorGood) : A.Split exteriorGood := by | |
| have h₁ := T.positive | |
| have h₂ := T.ordered | |
| have h₃ := T.inside | |
| have ho := T.odd₁ | |
| have he := T.even₂ | |
| have ha₁ : A.Admissible 0 (s - T.p₂) := by | |
| have h := A.admissible_symm T.third | |
| simpa only [reverse, Nat.sub_self] using h | |
| have ha₂ : A.Admissible (s - T.p₂) (s - T.p₁) := A.admissible_symm T.middle | |
| have ha₃ : A.Admissible (s - T.p₁) s := by | |
| have h := A.admissible_symm T.first | |
| simpa only [reverse, Nat.sub_zero] using h | |
| have g₁ : A.reverse.Good T.p₂ s → A.Good 0 (s - T.p₂) := by | |
| intro h | |
| simpa only [Nat.sub_self] using A.good_reverse h₃ le_rfl h | |
| have g₂ : A.reverse.Good T.p₁ T.p₂ → A.Good (s - T.p₂) (s - T.p₁) := | |
| A.good_reverse h₂ (by omega) | |
| have g₃ : A.reverse.Good 0 T.p₁ → A.Good (s - T.p₁) s := by | |
| intro h | |
| simpa only [Nat.sub_zero] using A.good_reverse h₁ (by omega) h | |
| refine ⟨s - T.p₂, s - T.p₁, by omega, by omega, by omega, by omega, by omega, | |
| ha₁, ha₂, ha₃, ?_, ?_, ?_⟩ | |
| · rcases T.some_good with ⟨hf, ht⟩ | ⟨hm, hg⟩ | |
| · exact Or.inl ⟨g₁ ht, g₃ hf⟩ | |
| · exact Or.inr ⟨g₂ hm, hg⟩ | |
| · intro hf ht | |
| obtain ⟨hm, hg⟩ := T.outer_long (by omega) (by omega) | |
| exact ⟨g₂ hm, hg⟩ | |
| · intro hm | |
| obtain ⟨hf, ht⟩ := T.middle_long (by omega) | |
| exact ⟨g₁ ht, g₃ hf⟩ | |
| theorem split (hG : G.IsAcyclic) (exteriorGood : Prop) | |
| (hs : 3 ≤ s) (hsodd : s % 2 = 1) (P : L) | |
| (hP₀ : P ∈ A.sets 0) (hPₛ : P ∈ A.sets s) | |
| (hbad : ¬exteriorGood → A.labels 0 = P ∨ A.labels (s - 1) = P) : | |
| Nonempty (A.Split exteriorGood) := by | |
| classical | |
| by_cases hext : exteriorGood | |
| · exact A.split_good_exterior hG hext hs hsodd P hP₀ hPₛ | |
| rcases hbad hext with hleft | hright | |
| · exact A.split_left_boundary hG exteriorGood hs hsodd P hP₀ hPₛ hleft | |
| · have hP₀' : P ∈ A.reverse.sets 0 := by simpa only [reverse, Nat.sub_zero] using hPₛ | |
| have hPₛ' : P ∈ A.reverse.sets s := by simpa only [reverse, Nat.sub_self] using hP₀ | |
| have hzero : A.reverse.labels 0 = P := by simpa only [reverse, Nat.sub_zero] using hright | |
| obtain ⟨T⟩ := A.reverse.split_left_boundary hG exteriorGood hs hsodd P hP₀' hPₛ' hzero | |
| exact ⟨T.unreverse A hsodd⟩ | |
| end ArcData | |
| structure CycleData (G : SimpleGraph L) (m : ℕ) where | |
| labels : ℤ → L | |
| periodic : ∀ i, labels (i + m) = labels i | |
| adjacent : ∀ i, labels (i - 1) = labels i ∨ G.Adj (labels (i - 1)) (labels i) | |
| end MatchingFPRAS.Cells | |
| end OAI | |