| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Graphs.Sets |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.Transport |
| variable {V E L : Type*} |
| namespace IndexedCycle |
|
|
| variable {bags : V → Finset L} |
|
|
| variable (c : IndexedCycle (BagEdge.Incident (bags := bags))) |
|
|
| variable {G : SimpleGraph L} |
|
|
| variable (hbags : ∀ v, ∀ a ∈ bags v, ∀ b ∈ bags v, a = b ∨ G.Adj a b) |
|
|
| lemma current_color_mem (i : ℤ) : (c.edgeAtZ i).color ∈ bags (c.vertexAtZ i) := |
| BagEdge.color_mem _ ((c.edgeAtZ_incidence i _).mpr (Or.inl rfl)) |
|
|
| lemma previous_color_mem (i : ℤ) : (c.edgeAtZ (i-1)).color ∈ bags (c.vertexAtZ i) := by |
| apply BagEdge.color_mem |
| apply (c.edgeAtZ_incidence (i-1) _).mpr |
| right |
| congr 1 |
| omega |
|
|
| def labelCycle : MatchingFPRAS.Cells.CycleData G c.size where |
| labels i := (c.edgeAtZ i).color |
| periodic i := by rw [c.edgeAtZ_periodic] |
| adjacent i := hbags (c.vertexAtZ i) _ (c.previous_color_mem i) _ (c.current_color_mem i) |
|
|
| variable [DecidableEq L] |
|
|
| lemma labelCycle_sets_subset (i : ℤ) : (c.labelCycle hbags).sets i ⊆ bags (c.vertexAtZ i) := by |
| intro l hl |
| simp only [MatchingFPRAS.Cells.CycleData.sets, labelCycle, |
| Finset.mem_insert, Finset.mem_singleton] at hl |
| rcases hl with rfl | rfl |
| · exact c.previous_color_mem i |
| · exact c.current_color_mem i |
|
|
| lemma admissible_common_bag {x y : ℕ} (h : (c.labelCycle hbags).Admissible x y) : |
| ∃ l, l ∈ bags (c.vertexAt x) ∧ l ∈ bags (c.vertexAt y) := by |
| obtain ⟨l, hlx, hly⟩ := h |
| refine ⟨l, ?_, ?_⟩ |
| · simpa [c.vertexAtZ_nat] using c.labelCycle_sets_subset hbags x hlx |
| · simpa [c.vertexAtZ_nat] using c.labelCycle_sets_subset hbags y hly |
|
|
| lemma labelSets_periodic (i : ℤ) : |
| (c.labelCycle hbags).sets (i+c.size) = (c.labelCycle hbags).sets i := by |
| have h : i + (c.size : ℤ) - 1 = (i-1) + c.size := by ring |
| simp only [MatchingFPRAS.Cells.CycleData.sets, labelCycle, h, c.edgeAtZ_periodic] |
|
|
| lemma edge_color_fixed (i : ℕ) : |
| (c.edgeAt i).color ∈ (c.labelCycle hbags).sets i ∧ |
| (c.edgeAt i).color ∈ (c.labelCycle hbags).sets (i+1) := by |
| have h : (c.edgeAtZ (i : ℤ)).color ∈ (c.labelCycle hbags).sets (i : ℤ) ∧ |
| (c.edgeAtZ (i : ℤ)).color ∈ (c.labelCycle hbags).sets ((i : ℤ)+1) := by |
| simp [MatchingFPRAS.Cells.CycleData.sets, labelCycle] |
| simpa only [c.edgeAtZ_nat] using h |
|
|
| noncomputable def chord (x y : ℕ) (hxy : x < y) (hy : y < c.size) |
| (ha : (c.labelCycle hbags).Admissible x y) : BagEdge bags := by |
| classical |
| exact if hshort : y = x+1 then c.edgeAt x |
| else if hwrap : x = 0 ∧ y+1 = c.size then c.edgeAt y |
| else |
| let l := ha.choose |
| BagEdge.ofEndpoints (c.vertexAt x) (c.vertexAt y) |
| (fun h => by |
| have := (c.vertexAt_eq_iff (a := 0) (by have := c.four_le; omega) |
| ⟨by omega, by omega⟩ ⟨by omega, by omega⟩).mp h |
| omega) l |
| (by simpa [c.vertexAtZ_nat] using c.labelCycle_sets_subset hbags x ha.choose_spec.1) |
| (by simpa [c.vertexAtZ_nat] using c.labelCycle_sets_subset hbags y ha.choose_spec.2) |
| |
| lemma chord_incidence (x y : ℕ) (hxy : x < y) (hy : y < c.size) |
| (ha : (c.labelCycle hbags).Admissible x y) (v : V) : |
| (c.chord hbags x y hxy hy ha).Incident v ↔ v = c.vertexAt x ∨ v = c.vertexAt y := by |
| classical |
| dsimp only [chord] |
| split_ifs with hshort hwrap |
| · simpa [hshort] using c.edgeAt_incidence x v |
| · rw [c.edgeAt_incidence] |
| have he : c.vertexAt (y+1) = c.vertexAt x := by |
| rw [hwrap.2, hwrap.1] |
| simpa using c.vertexAt_periodic 0 |
| rw [he] |
| tauto |
| · exact BagEdge.ofEndpoints_incidence _ _ _ _ _ _ _ |
|
|
| lemma chord_short (x y : ℕ) (hxy : x < y) (hy : y < c.size) |
| (ha : (c.labelCycle hbags).Admissible x y) (h : y = x+1) : |
| c.chord hbags x y hxy hy ha = c.edgeAt x := by |
| simp only [chord, dite_eq_left h] |
|
|
| lemma chord_wrap (x y : ℕ) (hxy : x < y) (hy : y < c.size) |
| (ha : (c.labelCycle hbags).Admissible x y) (h : x = 0 ∧ y+1 = c.size) : |
| c.chord hbags x y hxy hy ha = c.edgeAt y := by |
| have hn : y ≠ x+1 := by have := c.four_le; omega |
| simp only [chord, dite_eq_right hn, dite_eq_left h] |
| |
| lemma chord_color_fixed (x y : ℕ) (hxy : x < y) (hy : y < c.size) |
| (ha : (c.labelCycle hbags).Admissible x y) : |
| (c.chord hbags x y hxy hy ha).color ∈ (c.labelCycle hbags).sets x ∧ |
| (c.chord hbags x y hxy hy ha).color ∈ (c.labelCycle hbags).sets y := by |
| classical |
| dsimp only [chord] |
| split_ifs with hs hw |
| · simpa only [hs, Nat.cast_add, Nat.cast_one] using c.edge_color_fixed hbags x |
| · have hb := c.edge_color_fixed hbags y |
| refine ⟨?_, hb.1⟩ |
| have he : (y : ℤ)+1 = (0 : ℤ)+c.size := by omega |
| rw [he, c.labelSets_periodic hbags] at hb |
| simpa only [hw.1, Nat.cast_zero] using hb.2 |
| · exact ha.choose_spec |
|
|
| def cellCorners {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) : c.Corners where |
| a₀ := x |
| a₁ := p |
| a₂ := q |
| a₃ := y |
| lt₀ := hc.first_pos |
| lt₁ := hc.middle_pos |
| lt₂ := hc.third_pos |
| lt₃ := hy |
| odd₀ := hc.odd_first |
| odd₁ := hc.odd_middle |
| odd₂ := hc.odd_third |
| odd₃ := by |
| have he : c.size + x - y = c.size - (y-x) := by have := hc.first_pos; have := hc.middle_pos; have := hc.third_pos; omega |
| rw [he] |
| exact hc.odd_exterior |
| |
| noncomputable def cellChords {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) : Fin 4 → BagEdge bags := |
| ![c.chord hbags x p hc.first_pos (by have := hc.middle_pos; have := hc.third_pos; omega) hc.first, |
| c.chord hbags p q hc.middle_pos (by have := hc.third_pos; omega) hc.middle, |
| c.chord hbags q y hc.third_pos hy hc.third, |
| c.chord hbags x y (by have := hc.first_pos; have := hc.middle_pos; have := hc.third_pos; omega) |
| hy (by obtain ⟨l, hly, hlx⟩ := hc.exterior; exact ⟨l,hlx,hly⟩)] |
| |
| lemma cellChords_incidence {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (i : Fin 4) (v : V) : |
| (c.cellChords hbags hc hy i).Incident v ↔ |
| v = ((c.cellCorners hbags hc hy).side i).vertex 0 ∨ |
| v = ((c.cellCorners hbags hc hy).side i).vertex |
| ((c.cellCorners hbags hc hy).side i).length := by |
| simp only [Corners.side_first, Corners.side_last] |
| have hper : c.vertexAt (c.size+x) = c.vertexAt x := by |
| simpa only [Nat.add_comm] using c.vertexAt_periodic x |
| fin_cases i <;> simp [cellChords, chord_incidence, cellCorners, Corners.start, |
| Corners.stop, hper, or_comm] |
| |
| noncomputable def pathsOfCell [DecidableEq V] {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) : CellPaths (BagEdge.Incident (bags := bags)) := |
| (c.cellCorners hbags hc hy).cellPaths (c.cellChords hbags hc hy) |
| (c.cellChords_incidence hbags hc hy) |
|
|
| def sideGood (k : c.Corners) (i : Fin 4) : Prop := |
| k.length i = 1 ∨ ∃ l, l ∈ (c.labelCycle hbags).sets (k.start i+1) ∧ |
| l ∈ (c.labelCycle hbags).sets ((k.stop i : ℤ)-1) |
|
|
| lemma cell_sideGood {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (i : Fin 4) : |
| c.sideGood hbags (c.cellCorners hbags hc hy) i ↔ |
| ![(c.labelCycle hbags).Good x p, (c.labelCycle hbags).Good p q, |
| (c.labelCycle hbags).Good q y, (c.labelCycle hbags).ExteriorGood x y] i := by |
| have hxp := hc.first_pos |
| have hpq := hc.middle_pos |
| have hqy := hc.third_pos |
| have cast_sub (a : ℕ) (ha : 1 ≤ a) : ((a-1 : ℕ) : ℤ) = (a : ℤ)-1 := by omega |
| have he : ((c.size : ℤ)+(x : ℤ))-1 = ((x : ℤ)-1) + c.size := by ring |
| fin_cases i <;> simp only [sideGood, cellCorners, Corners.length, Corners.start, |
| Corners.stop] |
| · simp [MatchingFPRAS.Cells.CycleData.Good, MatchingFPRAS.Cells.CycleData.Admissible, |
| cast_sub p (by omega), Nat.cast_add, Nat.cast_one, show p-x=1 ↔ p=x+1 by omega] |
| · simp [MatchingFPRAS.Cells.CycleData.Good, MatchingFPRAS.Cells.CycleData.Admissible, |
| cast_sub q (by omega), Nat.cast_add, Nat.cast_one, show q-p=1 ↔ q=p+1 by omega] |
| · simp [MatchingFPRAS.Cells.CycleData.Good, MatchingFPRAS.Cells.CycleData.Admissible, |
| cast_sub y (by omega), Nat.cast_add, Nat.cast_one, show y-q=1 ↔ y=q+1 by omega] |
| · simp [MatchingFPRAS.Cells.CycleData.ExteriorGood, Nat.cast_add, he, |
| c.labelSets_periodic hbags, show c.size+x-y=1 ↔ y+1=x+c.size by omega] |
| |
| noncomputable def leafRepair (k : c.Corners) (i : Fin 4) : BagEdge bags := by |
| classical |
| exact if hg : c.sideGood hbags k i ∧ k.length i ≠ 1 then |
| let hl := hg.1.resolve_left hg.2 |
| BagEdge.ofEndpoints ((k.side i).vertex 1) ((k.side i).vertex ((k.side i).length-1)) |
| (by |
| have hp := (k.side i).positive |
| have ho := (k.side i).odd |
| have hn : (k.side i).length ≠ 1 := hg.2 |
| intro he |
| have hh := (k.side i).simple ⟨by omega, by omega⟩ ⟨by omega, by omega⟩ he |
| omega) |
| hl.choose |
| (by |
| have hm := c.labelCycle_sets_subset hbags (k.start i+1) hl.choose_spec.1 |
| have hv : c.vertexAtZ ((k.start i : ℤ)+1) = (k.side i).vertex 1 := by |
| simpa only [Corners.side_vertex, Nat.cast_add, Nat.cast_one] using c.vertexAtZ_nat (k.start i+1) |
| exact (congrArg (fun v => hl.choose ∈ bags v) hv).mp hm) |
| (by |
| have hm := c.labelCycle_sets_subset hbags ((k.stop i : ℤ)-1) hl.choose_spec.2 |
| have hb := k.bounds i |
| have he : ((k.stop i : ℤ)-1) = ((k.start i + ((k.side i).length-1) : ℕ) : ℤ) := by |
| rw [Corners.side_length] |
| dsimp [Corners.length] |
| omega |
| have hv : c.vertexAtZ ((k.stop i : ℤ)-1) = (k.side i).vertex ((k.side i).length-1) := by |
| rw [he, c.vertexAtZ_nat] |
| rfl |
| exact (congrArg (fun v => hl.choose ∈ bags v) hv).mp hm) |
| else c.edgeAt (k.start i) |
|
|
| lemma leafRepair_incidence (k : c.Corners) (i : Fin 4) |
| (hg : c.sideGood hbags k i) (hl : k.length i ≠ 1) (v : V) : |
| (c.leafRepair hbags k i).Incident v ↔ |
| v = (k.side i).vertex 1 ∨ v = (k.side i).vertex ((k.side i).length-1) := by |
| simp only [leafRepair, dite_eq_left (And.intro hg hl), BagEdge.ofEndpoints_incidence] |
|
|
| lemma leafRepair_color_fixed (k : c.Corners) (i : Fin 4) |
| (hg : c.sideGood hbags k i) (hl : k.length i ≠ 1) : |
| (c.leafRepair hbags k i).color ∈ (c.labelCycle hbags).sets (k.start i+1) ∧ |
| (c.leafRepair hbags k i).color ∈ (c.labelCycle hbags).sets ((k.stop i : ℤ)-1) := by |
| simp only [leafRepair, dite_eq_left (And.intro hg hl), BagEdge.ofEndpoints_color] |
| exact (hg.resolve_left hl).choose_spec |
|
|
| lemma cell_has_good_pair {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) : |
| ∃ b : Bool, ∀ i : Fin 4, CellPaths.sideBit i ≠ b → |
| c.sideGood hbags (c.cellCorners hbags hc hy) i := by |
| rcases hc.some_good with ⟨hf,ht⟩ | ⟨hm,he⟩ |
| · refine ⟨false, ?_⟩ |
| intro i hi |
| rw [c.cell_sideGood hbags hc hy] |
| fin_cases i <;> simp_all [CellPaths.sideBit] |
| · refine ⟨true, ?_⟩ |
| intro i hi |
| rw [c.cell_sideGood hbags hc hy] |
| fin_cases i <;> simp_all [CellPaths.sideBit] |
| |
| lemma cell_actual_repairs [DecidableEq V] {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (b : Bool) |
| (hg : ∀ i : Fin 4, CellPaths.sideBit i ≠ b → |
| c.sideGood hbags (c.cellCorners hbags hc hy) i) : |
| (c.pathsOfCell hbags hc hy).Repairs b |
| (c.leafRepair hbags (c.cellCorners hbags hc hy)) := by |
| intro i hi hl v |
| exact c.leafRepair_incidence hbags _ i (hg i hi) hl v |
|
|
| lemma cell_long_opposite_good {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (b : Bool) |
| (hl : ∀ i : Fin 4, CellPaths.sideBit i = b → |
| (c.cellCorners hbags hc hy).length i ≠ 1) : |
| ∀ i : Fin 4, CellPaths.sideBit i ≠ b → |
| c.sideGood hbags (c.cellCorners hbags hc hy) i := by |
| have hxp := hc.first_pos |
| have hpq := hc.middle_pos |
| have hqy := hc.third_pos |
| cases b |
| · have h₁ := hl 1 (by decide) |
| have h₃ := hl 3 (by decide) |
| simp only [Corners.length, Corners.start, Corners.stop, cellCorners] at h₁ h₃ |
| have hg := hc.middle_exterior_long (by dsimp at h₁; omega) (by dsimp at h₃; omega) |
| intro i hi |
| rw [c.cell_sideGood hbags hc hy] |
| fin_cases i <;> simp_all [CellPaths.sideBit] |
| · have h₀ := hl 0 (by decide) |
| have h₂ := hl 2 (by decide) |
| simp only [Corners.length, Corners.start, Corners.stop, cellCorners] at h₀ h₂ |
| have hg := hc.outer_long (by dsimp at h₀; omega) (by dsimp at h₂; omega) |
| intro i hi |
| rw [c.cell_sideGood hbags hc hy] |
| fin_cases i <;> simp_all [CellPaths.sideBit] |
| |
| lemma cellChords_fixed {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (i : Fin 4) : |
| (c.cellChords hbags hc hy i).color ∈ |
| (c.labelCycle hbags).sets ((c.cellCorners hbags hc hy).start i) ∧ |
| (c.cellChords hbags hc hy i).color ∈ |
| (c.labelCycle hbags).sets ((c.cellCorners hbags hc hy).stop i) := by |
| fin_cases i |
| · exact c.chord_color_fixed hbags x p hc.first_pos |
| (by have := hc.middle_pos; have := hc.third_pos; omega) hc.first |
| · exact c.chord_color_fixed hbags p q hc.middle_pos |
| (by have := hc.third_pos; omega) hc.middle |
| · exact c.chord_color_fixed hbags q y hc.third_pos hy hc.third |
| · have hb := c.chord_color_fixed hbags x y |
| (by have := hc.first_pos; have := hc.middle_pos; have := hc.third_pos; omega) hy |
| (by obtain ⟨l,hly,hlx⟩ := hc.exterior; exact ⟨l,hlx,hly⟩) |
| change _ ∈ (c.labelCycle hbags).sets y ∧ _ ∈ (c.labelCycle hbags).sets (c.size+x) |
| rw [add_comm, c.labelSets_periodic hbags] |
| exact ⟨hb.2,hb.1⟩ |
| |
| lemma cellChords_previous_fixed {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (i : Fin 4) : |
| (c.cellChords hbags hc hy (i+1)).color ∈ |
| (c.labelCycle hbags).sets ((c.cellCorners hbags hc hy).stop i) := by |
| have h := (c.cellChords_fixed hbags hc hy (i+1)).1 |
| fin_cases i |
| · exact h |
| · exact h |
| · exact h |
| · change _ ∈ (c.labelCycle hbags).sets (c.size+x) |
| rw [add_comm, c.labelSets_periodic hbags] |
| exact h |
|
|
| variable (height : L → ℝ) (hheight : ∀ l, 0 < height l) |
|
|
| variable (hstep : ∀ a b, G.Adj a b → height a ≤ 2 * height b) |
|
|
| include hheight hstep in |
| lemma fixed_height_ratio (i : ℤ) {a b : L} |
| (ha : a ∈ (c.labelCycle hbags).sets i) (hb : b ∈ (c.labelCycle hbags).sets i) : |
| height a ≤ 2 * height b := by |
| rcases (c.labelCycle hbags).sets_clique i a ha b hb with rfl | hab |
| · linarith [hheight a] |
| · exact hstep a b hab |
|
|
| include hheight hstep in |
| lemma cellChords_height_ratio {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (i j : Fin 4) : |
| height (c.cellChords hbags hc hy i).color ≤ 4 * height (c.cellChords hbags hc hy j).color := by |
| have hn (i : Fin 4) : |
| height (c.cellChords hbags hc hy i).color ≤ 2 * height (c.cellChords hbags hc hy (i+1)).color ∧ |
| height (c.cellChords hbags hc hy (i+1)).color ≤ 2 * height (c.cellChords hbags hc hy i).color := by |
| exact ⟨c.fixed_height_ratio hbags height hheight hstep _ |
| (c.cellChords_fixed hbags hc hy i).2 (c.cellChords_previous_fixed hbags hc hy i), |
| c.fixed_height_ratio hbags height hheight hstep _ |
| (c.cellChords_previous_fixed hbags hc hy i) (c.cellChords_fixed hbags hc hy i).2⟩ |
| have h₀ := hn 0 |
| have h₁ := hn 1 |
| have h₂ := hn 2 |
| have h₃ := hn 3 |
| change height (c.cellChords hbags hc hy 0).color ≤ 2 * height (c.cellChords hbags hc hy 1).color ∧ |
| height (c.cellChords hbags hc hy 1).color ≤ 2 * height (c.cellChords hbags hc hy 0).color at h₀ |
| change height (c.cellChords hbags hc hy 1).color ≤ 2 * height (c.cellChords hbags hc hy 2).color ∧ |
| height (c.cellChords hbags hc hy 2).color ≤ 2 * height (c.cellChords hbags hc hy 1).color at h₁ |
| change height (c.cellChords hbags hc hy 2).color ≤ 2 * height (c.cellChords hbags hc hy 3).color ∧ |
| height (c.cellChords hbags hc hy 3).color ≤ 2 * height (c.cellChords hbags hc hy 2).color at h₂ |
| change height (c.cellChords hbags hc hy 3).color ≤ 2 * height (c.cellChords hbags hc hy 0).color ∧ |
| height (c.cellChords hbags hc hy 0).color ≤ 2 * height (c.cellChords hbags hc hy 3).color at h₃ |
| have hp₀ := hheight (c.cellChords hbags hc hy 0).color |
| have hp₁ := hheight (c.cellChords hbags hc hy 1).color |
| have hp₂ := hheight (c.cellChords hbags hc hy 2).color |
| have hp₃ := hheight (c.cellChords hbags hc hy 3).color |
| rcases h₀ with ⟨h₀,h₀'⟩ |
| rcases h₁ with ⟨h₁,h₁'⟩ |
| rcases h₂ with ⟨h₂,h₂'⟩ |
| rcases h₃ with ⟨h₃,h₃'⟩ |
| have hi : i = 0 ∨ i = 1 ∨ i = 2 ∨ i = 3 := by omega |
| have hj : j = 0 ∨ j = 1 ∨ j = 2 ∨ j = 3 := by omega |
| rcases hi with rfl | rfl | rfl | rfl <;> |
| rcases hj with rfl | rfl | rfl | rfl <;> linarith |
| |
| def LocalEdge {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (e : BagEdge bags) : Prop := |
| ∃ i : Fin 4, e = c.cellChords hbags hc hy i ∨ |
| e = ((c.cellCorners hbags hc hy).side i).edge 0 ∨ |
| e = ((c.cellCorners hbags hc hy).side i).edge |
| (((c.cellCorners hbags hc hy).side i).length-1) ∨ |
| (c.sideGood hbags (c.cellCorners hbags hc hy) i ∧ |
| (c.cellCorners hbags hc hy).length i ≠ 1 ∧ |
| e = c.leafRepair hbags (c.cellCorners hbags hc hy) i) |
|
|
| lemma side_boundaries_fixed (k : c.Corners) (i : Fin 4) : |
| ((k.side i).edge 0).color ∈ (c.labelCycle hbags).sets (k.start i) ∧ |
| ((k.side i).edge 0).color ∈ (c.labelCycle hbags).sets (k.start i+1) ∧ |
| ((k.side i).edge ((k.side i).length-1)).color ∈ (c.labelCycle hbags).sets ((k.stop i : ℤ)-1) ∧ |
| ((k.side i).edge ((k.side i).length-1)).color ∈ (c.labelCycle hbags).sets (k.stop i) := by |
| have hb := k.bounds i |
| have he : k.start i + (k.length i-1) = k.stop i-1 := by dsimp [Corners.length]; omega |
| change (c.edgeAt (k.start i+0)).color ∈ _ ∧ (c.edgeAt (k.start i+0)).color ∈ _ ∧ |
| (c.edgeAt (k.start i + (k.length i-1))).color ∈ _ ∧ |
| (c.edgeAt (k.start i + (k.length i-1))).color ∈ _ |
| rw [Nat.add_zero, he] |
| have hs := c.edge_color_fixed hbags (k.start i) |
| have ht := c.edge_color_fixed hbags (k.stop i-1) |
| have hcast : ((k.stop i-1 : ℕ) : ℤ) = (k.stop i : ℤ)-1 := by omega |
| have hcast' : ((k.stop i-1 : ℕ) : ℤ)+1 = (k.stop i : ℤ) := by omega |
| exact ⟨hs.1, hs.2, by simpa only [hcast] using ht.1, by simpa only [hcast'] using ht.2⟩ |
|
|
| include hheight hstep in |
| lemma localEdge_height_ratio {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) {e : BagEdge bags} (he : c.LocalEdge hbags hc hy e) : |
| ∃ i : Fin 4, |
| height e.color ≤ 4 * height (c.cellChords hbags hc hy i).color ∧ |
| height (c.cellChords hbags hc hy i).color ≤ 4 * height e.color := by |
| obtain ⟨i,he⟩ := he |
| refine ⟨i,?_⟩ |
| have hh := hheight e.color |
| have hch := hheight (c.cellChords hbags hc hy i).color |
| have hbc := c.side_boundaries_fixed hbags (c.cellCorners hbags hc hy) i |
| have hc' := c.cellChords_fixed hbags hc hy i |
| rcases he with rfl | rfl | rfl | ⟨hg,hl,rfl⟩ |
| · constructor <;> linarith |
| · have h₁ := c.fixed_height_ratio hbags height hheight hstep _ hbc.1 hc'.1 |
| have h₂ := c.fixed_height_ratio hbags height hheight hstep _ hc'.1 hbc.1 |
| constructor <;> linarith |
| · have h₁ := c.fixed_height_ratio hbags height hheight hstep _ hbc.2.2.2 hc'.2 |
| have h₂ := c.fixed_height_ratio hbags height hheight hstep _ hc'.2 hbc.2.2.2 |
| constructor <;> linarith |
| · have hrc := c.leafRepair_color_fixed hbags _ i hg hl |
| have h₁ := c.fixed_height_ratio hbags height hheight hstep _ hrc.1 hbc.2.1 |
| have h₂ := c.fixed_height_ratio hbags height hheight hstep _ hbc.2.1 hrc.1 |
| have h₃ := c.fixed_height_ratio hbags height hheight hstep _ hbc.1 hc'.1 |
| have h₄ := c.fixed_height_ratio hbags height hheight hstep _ hc'.1 hbc.1 |
| constructor <;> linarith |
| |
| include hheight hstep in |
| |
| theorem cell_comparability {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (D : ℝ) (hD : 1 ≤ D) (w : BagEdge bags → ℝ) |
| (hw : ∀ e, height e.color / D ≤ w e ∧ w e ≤ 3 * height e.color) |
| {e f : BagEdge bags} (he : c.LocalEdge hbags hc hy e) (hf : c.LocalEdge hbags hc hy f) : |
| w e ≤ (1000 * D) * w f := by |
| obtain ⟨i,hei,hie⟩ := c.localEdge_height_ratio hbags height hheight hstep hc hy he |
| obtain ⟨j,hfj,hjf⟩ := c.localEdge_height_ratio hbags height hheight hstep hc hy hf |
| have hij := c.cellChords_height_ratio hbags height hheight hstep hc hy i j |
| have hh : height e.color ≤ 64 * height f.color := by linarith |
| have hDpos : 0 < D := by linarith |
| have hfpos : 0 < w f := lt_of_lt_of_le (div_pos (hheight f.color) hDpos) (hw f).1 |
| have hlo : height f.color ≤ w f * D := (div_le_iff₀ hDpos).mp (hw f).1 |
| have h192 : w e ≤ 192 * D * w f := by nlinarith [(hw e).2] |
| nlinarith [mul_pos hDpos hfpos] |
|
|
| lemma common_clamp_ratio {lo hi A x y : ℝ} (hlo : 0 ≤ lo) (hA : 1 ≤ A) |
| (hxy : x ≤ A * y) : |
| max lo (min hi x) ≤ A * max lo (min hi y) := by |
| have hnonneg : 0 ≤ max lo (min hi y) := le_trans hlo (le_max_left _ _) |
| have hupper : max lo (min hi y) ≤ A * max lo (min hi y) := by nlinarith |
| apply max_le |
| · exact le_trans (le_max_left _ _) hupper |
| · by_cases hy : y ≤ hi |
| · have hm : y ≤ max lo (min hi y) := by rw [min_eq_right hy]; exact le_max_right _ _ |
| exact le_trans (min_le_right _ _) (le_trans hxy (mul_le_mul_of_nonneg_left hm (by linarith))) |
| · have hm : hi ≤ max lo (min hi y) := by rw [min_eq_left (le_of_not_ge hy)]; exact le_max_right _ _ |
| exact le_trans (min_le_left _ _) (le_trans hm hupper) |
| |
| include hheight hstep in |
| lemma cell_clamp_comparability {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (D : ℝ) (hD : 1 ≤ D) (w : BagEdge bags → ℝ) |
| (hw : ∀ e, height e.color / D ≤ w e ∧ w e ≤ 3 * height e.color) |
| (lo hi : ℝ) (hlo : 0 ≤ lo) |
| {e f : BagEdge bags} (he : c.LocalEdge hbags hc hy e) (hf : c.LocalEdge hbags hc hy f) : |
| max lo (min hi (w e)) ≤ (1000 * D) * max lo (min hi (w f)) := |
| common_clamp_ratio hlo (by linarith) (c.cell_comparability hbags height hheight hstep hc hy D hD w hw he hf) |
|
|
| lemma drops_local [DecidableEq V] {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (r : Bool) {e : BagEdge bags} |
| (he : e ∈ (c.pathsOfCell hbags hc hy).drops r) : c.LocalEdge hbags hc hy e := by |
| obtain ⟨i, _, hi⟩ := Multiset.mem_sum.mp he |
| by_cases h : CellPaths.sideBit i = r |
| · simp only [ite_eq_left h, Multiset.notMem_zero] at hi |
| · simp only [ite_eq_right h, Finset.mem_val] at hi |
| refine ⟨i, Or.inr ?_⟩ |
| change e ∈ ((c.cellCorners hbags hc hy).side i).boundary at hi |
| by_cases hl : ((c.cellCorners hbags hc hy).side i).length = 1 |
| · simp only [OddPath.boundary, ite_eq_left hl, Finset.mem_singleton] at hi |
| exact Or.inl hi |
| · simp only [OddPath.boundary, ite_eq_right hl, Finset.mem_insert, Finset.mem_singleton] at hi |
| exact hi.imp_right Or.inl |
|
|
| lemma additions_local [DecidableEq V] {x p q y : ℕ} (hc : (c.labelCycle hbags).Cell x p q y) |
| (hy : y < c.size) (b r : Bool) |
| (hg : ∀ i, CellPaths.sideBit i ≠ r → c.sideGood hbags (c.cellCorners hbags hc hy) i) |
| {e : BagEdge bags} (he : e ∈ (c.pathsOfCell hbags hc hy).additions b r |
| (c.leafRepair hbags (c.cellCorners hbags hc hy))) : c.LocalEdge hbags hc hy e := by |
| rcases Multiset.mem_add.mp he with he | he |
| · obtain ⟨i, _, hi⟩ := Multiset.mem_sum.mp he |
| by_cases h : CellPaths.sideBit i = b |
| · simp only [ite_eq_left h, Multiset.mem_singleton] at hi |
| exact ⟨i, Or.inl hi⟩ |
| · simp only [ite_eq_right h, Multiset.notMem_zero] at hi |
| · obtain ⟨i, _, hi⟩ := Multiset.mem_sum.mp he |
| by_cases h : CellPaths.sideBit i = r |
| · simp only [ite_eq_left h, Multiset.notMem_zero] at hi |
| · simp only [ite_eq_right h] at hi |
| change e ∈ ((c.cellCorners hbags hc hy).side i).repairAdded (c.leafRepair hbags (c.cellCorners hbags hc hy) i) at hi |
| by_cases hl : ((c.cellCorners hbags hc hy).side i).length = 1 |
| · simp only [OddPath.repairAdded, ite_eq_left hl, Multiset.notMem_zero] at hi |
| · simp only [OddPath.repairAdded, ite_eq_right hl, Multiset.mem_singleton] at hi |
| exact ⟨i, Or.inr (Or.inr (Or.inr ⟨hg i h, hl, hi⟩))⟩ |
|
|
| end IndexedCycle |
|
|
| end MatchingFPRAS.Transport |
|
|
| end OAI |
|
|
