| import OAI.Probability.LatticePassage.Geodesic | |
| namespace OAI | |
| namespace CompanionFPP | |
| private theorem horizontal_length (a b : ℤ) (n : ℕ) : | |
| ∃ p : lattice.Walk (a, b) (a + (n : ℤ), b), p.length = n := by | |
| induction n with | |
| | zero => simp | |
| | succ n ih => | |
| obtain ⟨p,hp⟩ := ih | |
| have h : lattice.Adj (a + (n : ℤ), b) (a + ((n+1 : ℕ) : ℤ), b) := by | |
| change |a + (n : ℤ) - (a + ((n+1 : ℕ) : ℤ))| + |b - b| = 1 | |
| simp | |
| exact ⟨p.concat h, by simp [hp]⟩ | |
| private theorem vertical_length (a b : ℤ) (n : ℕ) : | |
| ∃ p : lattice.Walk (a, b) (a, b + (n : ℤ)), p.length = n := by | |
| induction n with | |
| | zero => simp | |
| | succ n ih => | |
| obtain ⟨p,hp⟩ := ih | |
| have h : lattice.Adj (a, b + (n : ℤ)) (a, b + ((n+1 : ℕ) : ℤ)) := by | |
| change |a - a| + |b + (n : ℤ) - (b + ((n+1 : ℕ) : ℤ))| = 1 | |
| simp | |
| exact ⟨p.concat h, by simp [hp]⟩ | |
| theorem horizontal_coordinate_length (a c b : ℤ) : | |
| ∃ p : lattice.Walk (a,b) (c,b), p.length = (c-a).natAbs := by | |
| by_cases h : a ≤ c | |
| · have he : a + ((c-a).toNat : ℤ) = c := by omega | |
| have hn : (c-a).toNat = (c-a).natAbs := by omega | |
| have hh := horizontal_length a b (c-a).toNat | |
| rw [he] at hh | |
| simpa only [hn] using hh | |
| · have he : c + ((a-c).toNat : ℤ) = a := by omega | |
| obtain ⟨p,hp⟩ := horizontal_length c b (a-c).toNat | |
| have hn : (a-c).toNat = (c-a).natAbs := by omega | |
| have hq : ∃ p : lattice.Walk (c,b) (a,b), p.length = (c-a).natAbs := by | |
| have hh : ∃ p : lattice.Walk (c,b) (c + ((a-c).toNat : ℤ),b), | |
| p.length = (a-c).toNat := ⟨p,hp⟩ | |
| rw [he] at hh | |
| simpa only [hn] using hh | |
| obtain ⟨q,hq⟩ := hq | |
| exact ⟨q.reverse, by simpa using hq⟩ | |
| theorem vertical_coordinate_length (a b c : ℤ) : | |
| ∃ p : lattice.Walk (a,b) (a,c), p.length = (c-b).natAbs := by | |
| by_cases h : b ≤ c | |
| · have he : b + ((c-b).toNat : ℤ) = c := by omega | |
| have hn : (c-b).toNat = (c-b).natAbs := by omega | |
| have hh := vertical_length a b (c-b).toNat | |
| rw [he] at hh | |
| simpa only [hn] using hh | |
| · have he : c + ((b-c).toNat : ℤ) = b := by omega | |
| have hn : (b-c).toNat = (c-b).natAbs := by omega | |
| have hq : ∃ p : lattice.Walk (a,c) (a,b), p.length = (c-b).natAbs := by | |
| have hh := vertical_length a c (b-c).toNat | |
| rw [he] at hh | |
| simpa only [hn] using hh | |
| obtain ⟨q,hq⟩ := hq | |
| exact ⟨q.reverse, by simpa using hq⟩ | |
| def latticeDistance (x y : Vertex) : ℕ := (y.1-x.1).natAbs + (y.2-x.2).natAbs | |
| theorem coordinate_path (x y : Vertex) : | |
| ∃ p : lattice.Walk x y, p.IsPath ∧ p.length ≤ latticeDistance x y := by | |
| classical | |
| obtain ⟨p,hp⟩ := horizontal_coordinate_length x.1 y.1 x.2 | |
| obtain ⟨q,hq⟩ := vertical_coordinate_length y.1 x.2 y.2 | |
| refine ⟨(p.append q).bypass, (p.append q).bypass_isPath, ?_⟩ | |
| exact (p.append q).length_bypass_le_length.trans (by simp [hp, hq, latticeDistance]) | |
| end CompanionFPP | |
| end OAI | |