| import OAI.Combinatorics.Hamiltonian.Traces |
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| namespace OAI |
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| /-! Ordinary square cells and connected finite lattice corridors. -/ |
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| namespace Barnette.SquareGrid |
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| noncomputable section |
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| def graph : SimpleGraph (ℤ × ℤ) where |
| Adj a b := ((a.1 + 1 = b.1 ∨ b.1 + 1 = a.1) ∧ a.2 = b.2) ∨ |
| ((a.2 + 1 = b.2 ∨ b.2 + 1 = a.2) ∧ a.1 = b.1) |
| symm := ⟨by |
| intro a b h |
| rcases h with ⟨h, he⟩ | ⟨h, he⟩ |
| · exact Or.inl ⟨h.symm, he.symm⟩ |
| · exact Or.inr ⟨h.symm, he.symm⟩⟩ |
| loopless := ⟨by intro a h; rcases h with ⟨h, _⟩ | ⟨h, _⟩ <;> omega⟩ |
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| def cell (δ : ℝ) (z : ℤ × ℤ) : Set (ℝ × ℝ) := |
| Set.Icc (δ * z.1) (δ * ((z.1 : ℝ) + 1)) ×ˢ |
| Set.Icc (δ * z.2) (δ * ((z.2 : ℝ) + 1)) |
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| def cells (δ : ℝ) (K : Set (ℝ × ℝ)) : Set (ℤ × ℤ) := |
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| theorem cell_closed (δ : ℝ) (z : ℤ × ℤ) : IsClosed (cell δ z) := |
| isClosed_Icc.prod isClosed_Icc |
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| theorem cell_compact (δ : ℝ) (z : ℤ × ℤ) : IsCompact (cell δ z) := |
| isCompact_Icc.prod isCompact_Icc |
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| theorem cell_cover (hδ : 0 < δ) (p : ℝ × ℝ) : ∃ z, p ∈ cell δ z := by |
| refine ⟨(⌊p.1 / δ⌋, ⌊p.2 / δ⌋), ?_⟩ |
| constructor |
| · constructor |
| · have h := (le_div_iff₀ hδ).mp (Int.floor_le (p.1 / δ)) |
| simpa only [mul_comm] using h |
| · have h := (div_lt_iff₀ hδ).mp (Int.lt_floor_add_one (p.1 / δ)) |
| exact le_of_lt (by simpa only [mul_comm] using h) |
| · constructor |
| · have h := (le_div_iff₀ hδ).mp (Int.floor_le (p.2 / δ)) |
| simpa only [mul_comm] using h |
| · have h := (div_lt_iff₀ hδ).mp (Int.lt_floor_add_one (p.2 / δ)) |
| exact le_of_lt (by simpa only [mul_comm] using h) |
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| theorem subset_union_cells (hδ : 0 < δ) (K : Set (ℝ × ℝ)) : |
| K ⊆ ⋃ z : cells δ K, cell δ z.val := by |
| intro p hp |
| obtain ⟨z, hz⟩ := cell_cover hδ p |
| exact Set.mem_iUnion.mpr ⟨⟨z, p, hz, hp⟩, hz⟩ |
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| private theorem int_bound_of_cell (hδ : 0 < δ) |
| (hx : -R ≤ x ∧ x ≤ R) (ha : δ * a ≤ x ∧ x ≤ δ * ((a : ℝ) + 1)) : |
| -(R / δ + 1) ≤ (a : ℝ) ∧ (a : ℝ) ≤ R / δ + 1 := by |
| have hu : (a : ℝ) ≤ R / δ := |
| (le_div_iff₀ hδ).mpr (by nlinarith [ha.1, hx.2]) |
| have hl : -R / δ ≤ (a : ℝ) + 1 := |
| (div_le_iff₀ hδ).mpr (by nlinarith [ha.2, hx.1]) |
| rw [neg_div] at hl |
| constructor <;> linarith |
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| theorem cells_finite (hδ : 0 < δ) |
| (hK : IsCompact K) : (cells δ K).Finite := by |
| obtain ⟨R, _, hR⟩ := hK.isBounded.exists_pos_norm_le |
| obtain ⟨N, hN⟩ := exists_int_ge (R / δ + 1) |
| apply ((Set.finite_Icc (-N) N).prod (Set.finite_Icc (-N) N)).subset |
| rintro z ⟨p, hp, hpK⟩ |
| have hp₁ : -R ≤ p.1 ∧ p.1 ≤ R := abs_le.mp ((norm_fst_le p).trans (hR p hpK)) |
| have hp₂ : -R ≤ p.2 ∧ p.2 ≤ R := abs_le.mp ((norm_snd_le p).trans (hR p hpK)) |
| obtain ⟨hz₁, hz₁'⟩ := int_bound_of_cell hδ hp₁ hp.1 |
| obtain ⟨hz₂, hz₂'⟩ := int_bound_of_cell hδ hp₂ hp.2 |
| have hb₁ : (-(N : ℝ)) ≤ z.1 ∧ (z.1 : ℝ) ≤ N := by constructor <;> linarith |
| have hb₂ : (-(N : ℝ)) ≤ z.2 ∧ (z.2 : ℝ) ≤ N := by constructor <;> linarith |
| exact ⟨⟨by exact_mod_cast hb₁.1, by exact_mod_cast hb₁.2⟩, |
| ⟨by exact_mod_cast hb₂.1, by exact_mod_cast hb₂.2⟩⟩ |
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| theorem dist_le_of_mem_cell |
| (hp : p ∈ cell δ z) (hq : q ∈ cell δ z) : dist p q ≤ δ := by |
| rw [Prod.dist_eq, max_le_iff, Real.dist_eq, Real.dist_eq] |
| constructor <;> rw [abs_le] <;> constructor <;> |
| nlinarith [hp.1.1, hp.1.2, hp.2.1, hp.2.2, hq.1.1, hq.1.2, hq.2.1, hq.2.2] |
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| theorem cell_subset_thickening (hδε : δ < ε) |
| (hz : z ∈ cells δ K) : cell δ z ⊆ Metric.thickening ε K := by |
| obtain ⟨q, hq, hqK⟩ := hz |
| intro p hp |
| exact Metric.mem_thickening_iff.mpr ⟨q, hqK, (dist_le_of_mem_cell hp hq).trans_lt hδε⟩ |
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| theorem cells_disjoint (hδε : δ < ε) |
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| (hKL : Disjoint (Metric.thickening ε K) (Metric.thickening ε L)) : |
| Disjoint (cells δ K) (cells δ L) := by |
| rw [Set.disjoint_left] at hKL ⊢ |
| intro z hz hz' |
| have hzp := hz |
| obtain ⟨p, hp, _⟩ := hzp |
| exact hKL (cell_subset_thickening hδε hz hp) (cell_subset_thickening hδε hz' hp) |
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| end |
| end Barnette.SquareGrid |
| namespace Barnette.SquareGrid |
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| noncomputable section |
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| theorem finite_closed_cover_preconnected [Finite ι] [TopologicalSpace α] |
| (H : SimpleGraph ι) (K : Set α) (C : ι → Set α) |
| (hK : IsPreconnected K) (hC : ∀ i, IsClosed (C i)) |
| (hcover : K ⊆ ⋃ i, C i) (hne : ∀ i, (K ∩ C i).Nonempty) |
| (hjoin : ∀ p ∈ K, ∀ i j, p ∈ C i → p ∈ C j → H.Reachable i j) : |
| H.Preconnected := by |
| classical |
| intro a b |
| by_contra hab |
| let U : Set α := ⋃ i : , C i.val |
| let W : Set α := ⋃ i : , C i.val |
| have hU : IsClosed U := isClosed_iUnion_of_finite (fun i => hC i.val) |
| have hW : IsClosed W := isClosed_iUnion_of_finite (fun i => hC i.val) |
| have hUW : K ⊆ U ∪ W := by |
| intro p hp |
| obtain ⟨i, hi⟩ := Set.mem_iUnion.mp (hcover hp) |
| by_cases hai : H.Reachable a i |
| · exact Or.inl (Set.mem_iUnion.mpr ⟨⟨i, hai⟩, hi⟩) |
| · exact Or.inr (Set.mem_iUnion.mpr ⟨⟨i, hai⟩, hi⟩) |
| have haU : (K ∩ U).Nonempty := by |
| obtain ⟨p, hpK, hpa⟩ := hne a |
| exact ⟨p, hpK, Set.mem_iUnion.mpr ⟨⟨a, .rfl⟩, hpa⟩⟩ |
| have hbW : (K ∩ W).Nonempty := by |
| obtain ⟨p, hpK, hpb⟩ := hne b |
| exact ⟨p, hpK, Set.mem_iUnion.mpr ⟨⟨b, hab⟩, hpb⟩⟩ |
| obtain ⟨p, hpK, hpU, hpW⟩ := |
| isPreconnected_closed_iff.mp hK U W hU hW hUW haU hbW |
| obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hpU |
| obtain ⟨j, hj⟩ := Set.mem_iUnion.mp hpW |
| exact j.property (i.property.trans (hjoin p hpK i.val j.val hi hj)) |
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| private theorem coordinates_close (hδ : 0 < δ) |
| (ha : δ * a ≤ p ∧ p ≤ δ * ((a : ℝ) + 1)) |
| (hb : δ * b ≤ p ∧ p ≤ δ * ((b : ℝ) + 1)) : |
| a ≤ b + 1 ∧ b ≤ a + 1 := by |
| have hab : (a : ℝ) ≤ (b : ℝ) + 1 := (mul_le_mul_iff_right₀ hδ).mp (ha.1.trans hb.2) |
| have hba : (b : ℝ) ≤ (a : ℝ) + 1 := (mul_le_mul_iff_right₀ hδ).mp (hb.1.trans ha.2) |
| exact ⟨by exact_mod_cast hab, by exact_mod_cast hba⟩ |
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| theorem reachable_of_common_point (hδ : 0 < δ) |
| (hp : p ∈ K) (a b : cells δ K) |
| (ha : p ∈ cell δ a.val) (hb : p ∈ cell δ b.val) : |
| (graph.induce (cells δ K)).Reachable a b := by |
| obtain ⟨hx, hx'⟩ := coordinates_close hδ ha.1 hb.1 |
| obtain ⟨hy, hy'⟩ := coordinates_close hδ ha.2 hb.2 |
| let c : cells δ K := ⟨(b.val.1, a.val.2), p, ⟨hb.1, ha.2⟩, hp⟩ |
| have hac : (graph.induce (cells δ K)).Reachable a c := by |
| by_cases h : a.val.1 = b.val.1 |
| · have heq : a = c := Subtype.ext (Prod.ext h rfl) |
| rw [← heq] |
| · apply SimpleGraph.Adj.reachable |
| change graph.Adj a.val c.val |
| exact Or.inl ⟨by dsimp [c]; omega, rfl⟩ |
| have hcb : (graph.induce (cells δ K)).Reachable c b := by |
| by_cases h : a.val.2 = b.val.2 |
| · have heq : c = b := Subtype.ext (Prod.ext rfl h) |
| rw [heq] |
| · apply SimpleGraph.Adj.reachable |
| change graph.Adj c.val b.val |
| exact Or.inr ⟨by dsimp [c]; omega, rfl⟩ |
| exact hac.trans hcb |
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| theorem cells_connected (hδ : 0 < δ) |
| (hK : IsCompact K) (hc : IsConnected K) : |
| (graph.induce (cells δ K)).Connected := by |
| classical |
| let _ : Fintype (cells δ K) := (cells_finite hδ hK).fintype |
| have hone : Nonempty (cells δ K) := by |
| obtain ⟨p, hp⟩ := hc.nonempty |
| obtain ⟨z, hz⟩ := cell_cover hδ p |
| exact ⟨⟨z, p, hz, hp⟩⟩ |
| refine ⟨finite_closed_cover_preconnected _ K (fun z : cells δ K => cell δ z.val) |
| hc.isPreconnected (fun z => cell_closed _ _) (subset_union_cells hδ K) ?_ ?_⟩ |
| · intro z |
| obtain ⟨p, hp, hpK⟩ := z.property |
| exact ⟨p, hpK, hp⟩ |
| · intro p hp a b ha hb |
| exact reachable_of_common_point hδ hp a b ha hb |
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| end |
| end Barnette.SquareGrid |
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| end OAI |
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