| import OAI.Combinatorics.CycleDecomposition.CoupledResolution |
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| namespace OAI |
|
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| universe cycleUniverse1 cycleUniverse2 |
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| section |
| open Filter Asymptotics Real |
| open scoped Topology |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset |
| section |
|
|
| namespace ErdosGallai.Scale |
| universe u |
| open Finset SimpleGraph |
| variable {V : Type cycleUniverse1} [Fintype V] [DecidableEq V] |
|
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| structure DFSState (G : SimpleGraph V) where |
| stack : List V |
| finished : Finset V |
| unexplored : Finset V |
| nodup : stack.Nodup |
| chain : stack.IsChain G.Adj |
| f_s : Disjoint finished stack.toFinset |
| f_u : Disjoint finished unexplored |
| s_u : Disjoint stack.toFinset unexplored |
| cover : finished ∪ stack.toFinset ∪ unexplored = univ |
| separation : ∀ x ∈ finished, ∀ y ∈ unexplored, ¬ G.Adj x y |
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| namespace DFSState |
| variable {G : SimpleGraph V} |
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|
| lemma card_eq (s : DFSState G) : |
| Fintype.card V = s.finished.card + s.stack.length + s.unexplored.card := by |
| have hdis : Disjoint (s.finished ∪ s.stack.toFinset) s.unexplored := |
| Finset.disjoint_union_left.mpr ⟨s.f_u, s.s_u⟩ |
| have hc := congrArg Finset.card s.cover |
| rw [Finset.card_union_of_disjoint hdis, Finset.card_union_of_disjoint s.f_s, |
| List.toFinset_card_of_nodup s.nodup, Finset.card_univ] at hc |
| exact hc.symm |
|
|
| lemma restart (s : DFSState G) (hs : s.stack = []) (v : V) (hv : v ∈ s.unexplored) : |
| ∃ t : DFSState G, t.finished = s.finished ∧ |
| t.unexplored.card + 1 = s.unexplored.card := by |
| have hvf : v ∉ s.finished := fun hf => Finset.disjoint_left.mp s.f_u hf hv |
| refine ⟨⟨[v], s.finished, s.unexplored.erase v, by simp, by simp, |
| ?_, ?_, ?_, ?_, ?_⟩, rfl, ?_⟩ |
| · simpa using hvf |
| · exact s.f_u.mono_right (Finset.erase_subset _ _) |
| · simp |
| · change s.finished ∪ {v} ∪ s.unexplored.erase v = univ |
| rw [Finset.union_assoc, Finset.singleton_union, Finset.insert_erase hv] |
| simpa only [hs, List.toFinset_nil, Finset.union_empty] using s.cover |
| · intro x hx y hy |
| exact s.separation x hx y (Finset.mem_of_mem_erase hy) |
| · exact Finset.card_erase_add_one hv |
|
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| lemma push (s : DFSState G) (v w : V) (L : List V) (hs : s.stack = v::L) |
| (hw : w ∈ s.unexplored) (hadj : G.Adj w v) : |
| ∃ t : DFSState G, t.finished = s.finished ∧ |
| t.unexplored.card + 1 = s.unexplored.card := by |
| have hws : w ∉ s.stack := fun h => Finset.disjoint_left.mp s.s_u |
| (List.mem_toFinset.mpr h) hw |
| have hwf : w ∉ s.finished := fun h => Finset.disjoint_left.mp s.f_u h hw |
| refine ⟨⟨w::s.stack, s.finished, s.unexplored.erase w, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩, |
| rfl, ?_⟩ |
| · exact List.nodup_cons.mpr ⟨hws,s.nodup⟩ |
| · rw [hs] |
| exact List.IsChain.cons_cons hadj (hs ▸ s.chain) |
| · simpa using And.intro hwf s.f_s |
| · exact s.f_u.mono_right (Finset.erase_subset _ _) |
| · rw [List.toFinset_cons, Finset.disjoint_insert_left] |
| exact ⟨(by simp), s.s_u.mono_right (Finset.erase_subset _ _)⟩ |
| · ext x |
| have hx : x ∈ s.finished ∪ s.stack.toFinset ∪ s.unexplored := by rw [s.cover]; simp |
| simp only [Finset.mem_union] at hx |
| simp only [List.toFinset_cons, Finset.mem_union, Finset.mem_insert, |
| Finset.mem_erase, Finset.mem_univ, iff_true] |
| by_cases hxw : x = w |
| · exact Or.inl (Or.inr (Or.inl hxw)) |
| · rcases hx with (h|h)|h |
| · exact Or.inl (Or.inl h) |
| · exact Or.inl (Or.inr (Or.inr h)) |
| · exact Or.inr ⟨hxw,h⟩ |
| · intro x hx y hy |
| exact s.separation x hx y (Finset.mem_of_mem_erase hy) |
| · exact Finset.card_erase_add_one hw |
|
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| lemma pop (s : DFSState G) (v : V) (L : List V) (hs : s.stack = v::L) |
| (hn : ∀ w ∈ s.unexplored, ¬G.Adj v w) : |
| ∃ t : DFSState G, t.finished.card = s.finished.card + 1 ∧ |
| t.unexplored = s.unexplored := by |
| have hvf : v ∉ s.finished := fun hv => Finset.disjoint_left.mp s.f_s hv (by simp [hs]) |
| have hvu : v ∉ s.unexplored := fun hv => Finset.disjoint_left.mp s.s_u (by simp [hs]) hv |
| have hL : L.toFinset ⊆ s.stack.toFinset := by rw [hs]; simp |
| have hnd : v ∉ L ∧ L.Nodup := by simpa only [hs, List.nodup_cons] using s.nodup |
| have hvL : v ∉ L := hnd.1 |
| refine ⟨⟨L, insert v s.finished, s.unexplored, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩, ?_, rfl⟩ |
| · exact hnd.2 |
| · exact (hs ▸ s.chain).of_cons |
| · exact Finset.disjoint_insert_left.mpr ⟨by simpa using hvL, s.f_s.mono_right hL⟩ |
| · exact Finset.disjoint_insert_left.mpr ⟨hvu,s.f_u⟩ |
| · exact s.s_u.mono_left hL |
| · simpa [hs, Finset.insert_union] using s.cover |
| · intro x hx y hy |
| rcases Finset.mem_insert.mp hx with rfl|hx |
| · exact hn y hy |
| · exact s.separation x hx y hy |
| · exact Finset.card_insert_of_notMem hvf |
|
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| lemma reaches_balance (s : DFSState G) (hs : s.finished.card ≤ s.unexplored.card) : |
| ∃ t : DFSState G, t.finished.card = t.unexplored.card := by |
| generalize hd : s.unexplored.card - s.finished.card = d |
| induction d using Nat.strong_induction_on generalizing s with |
| | h d ih => |
| by_cases heq : s.finished.card = s.unexplored.card |
| · exact ⟨s,heq⟩ |
| have hlt : s.finished.card < s.unexplored.card := lt_of_le_of_ne hs heq |
| have hdpos : 0 < d := by omega |
| cases hstack : s.stack with |
| | nil => |
| obtain ⟨v,hv⟩ := Finset.card_pos.mp (show 0 < s.unexplored.card by omega) |
| obtain ⟨t,htf,htu⟩ := s.restart hstack v hv |
| apply ih (t.unexplored.card-t.finished.card) (by rw [htf]; omega) t (by rw [htf]; omega) rfl |
| | cons v L => |
| by_cases hn : ∃ w ∈ s.unexplored, G.Adj v w |
| · obtain ⟨w,hw,hadj⟩ := hn |
| obtain ⟨t,htf,htu⟩ := s.push v w L hstack hw hadj.symm |
| apply ih (t.unexplored.card-t.finished.card) (by rw [htf]; omega) t (by rw [htf]; omega) rfl |
| · push Not at hn |
| obtain ⟨t,htf,htu⟩ := s.pop v L hstack hn |
| apply ih (t.unexplored.card-t.finished.card) (by rw [htf,htu]; omega) t (by rw [htf,htu]; omega) rfl |
| |
| end DFSState |
| |
| theorem exists_balanced_dfs_state (G : SimpleGraph V) : |
| ∃ s : DFSState G, s.finished.card = s.unexplored.card := by |
| let s : DFSState G := ⟨[], ∅, univ, by simp, by simp, by simp, by simp, by simp, |
| by simp, by simp⟩ |
| exact s.reaches_balance (by simp [s]) |
|
|
| end ErdosGallai.Scale |
|
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| namespace ErdosGallai.Scale |
| open Finset SimpleGraph |
| variable {V : Type cycleUniverse2} [Fintype V] [DecidableEq V] |
|
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| def externalNeighbors (G : SimpleGraph V) [DecidableRel G.Adj] (U : Finset V) : Finset V := |
| univ.filter (fun v => v ∉ U ∧ ∃ u ∈ U, G.Adj u v) |
|
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| def VertexExpansion (G : SimpleGraph V) [DecidableRel G.Adj] (α : ℝ) : Prop := |
| ∀ U : Finset V, U.Nonempty → 2 * U.card ≤ Fintype.card V → |
| α * U.card ≤ ((externalNeighbors G U).card : ℝ) |
|
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| lemma DFSState.external_subset (G : SimpleGraph V) [DecidableRel G.Adj] (s : DFSState G) : |
| externalNeighbors G s.unexplored ⊆ s.stack.toFinset := by |
| intro v hv |
| obtain ⟨_,hvu,u,hu,hadj⟩ := Finset.mem_filter.mp hv |
| have hc : v ∈ s.finished ∪ s.stack.toFinset ∪ s.unexplored := by rw [s.cover]; simp |
| rcases Finset.mem_union.mp hc with h|h |
| · rcases Finset.mem_union.mp h with h|h |
| · exact (s.separation v h u hu hadj.symm).elim |
| · exact h |
| · exact (hvu h).elim |
| |
| theorem exists_long_stack (G : SimpleGraph V) [DecidableRel G.Adj] |
| (α : ℝ) (hα : α ≤ 1) (hexp : VertexExpansion G α) : |
| ∃ L : List V, L.Nodup ∧ L.IsChain G.Adj ∧ α * Fintype.card V / 3 ≤ (L.length : ℝ) := by |
| obtain ⟨s,heq⟩ := exists_balanced_dfs_state G |
| refine ⟨s.stack,s.nodup,s.chain,?_⟩ |
| have hc := s.card_eq |
| rw [heq] at hc |
| have hcR : (Fintype.card V : ℝ) = 2*(s.unexplored.card : ℝ)+(s.stack.length : ℝ) := by |
| exact_mod_cast (show Fintype.card V = 2*s.unexplored.card+s.stack.length by omega) |
| have hN : ((externalNeighbors G s.unexplored).card : ℝ) ≤ (s.stack.length : ℝ) := by |
| have ht := Finset.card_le_card (s.external_subset G) |
| rw [List.toFinset_card_of_nodup s.nodup] at ht |
| exact_mod_cast ht |
| by_cases hU : s.unexplored.Nonempty |
| · have hsmall : 2*s.unexplored.card ≤ Fintype.card V := by omega |
| have he := (hexp s.unexplored hU hsmall).trans hN |
| have ht : (0 : ℝ) ≤ s.stack.length := Nat.cast_nonneg _ |
| have ha := mul_le_mul_of_nonneg_right hα ht |
| nlinarith |
| · have hu : s.unexplored.card = 0 := by simp [Finset.not_nonempty_iff_eq_empty.mp hU] |
| rw [hu, Nat.cast_zero, mul_zero, zero_add] at hcR |
|
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| nlinarith [show (0 : ℝ) ≤ s.stack.length from Nat.cast_nonneg _, |
| mul_le_mul_of_nonneg_right hα (show (0 : ℝ) ≤ s.stack.length from Nat.cast_nonneg _)] |
|
|
| end ErdosGallai.Scale |
| end |
| end |
| end |
|
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| end OAI |
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