| import OAI.Combinatorics.MatchingPSD.CrossSelection |
|
|
| namespace OAI |
|
|
| namespace MatchingSmoothing |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma triple_pairing_five {s : ℕ} (hs : 1 ≤ s) (j : Fin 5) : |
| ∃ M : Pairing (TripleVertices s), crossVector tripleColor M = |
| (![![1,0,0], ![0,1,1], ![3,0,0], ![1,2,0], ![1,0,2]] j : Fin 3 → ℝ) := by |
| have hex (E : Fin 6 → ℕ) (he : endpointSizes E = ![2*s+1,2*s+1,2*s]) : |
| ∃ M : Pairing (TripleVertices s), crossVector tripleColor M = |
| (fun i => (![E 1,E 2,E 4] i : ℝ)) := by |
| obtain ⟨M,hM⟩ := exists_pairing_of_endpoints tripleColor E (by rw [tripleColor_histogram, he]) |
| exact ⟨M,funext hM⟩ |
| fin_cases j |
| · obtain ⟨M,hM⟩ := hex ![s,1,0,s,0,s] (by ext i |
| exact ⟨M,by rw [hM] |
| · obtain ⟨M,hM⟩ := hex ![s,0,1,s,1,s-1] (by ext i |
| exact ⟨M,by rw [hM] |
| · obtain ⟨M,hM⟩ := hex ![s-1,3,0,s-1,0,s] (by ext i |
| exact ⟨M,by rw [hM] |
| · obtain ⟨M,hM⟩ := hex ![s-1,1,2,s,0,s-1] (by ext i |
| exact ⟨M,by rw [hM] |
| · obtain ⟨M,hM⟩ := hex ![s,1,0,s-1,2,s-1] (by ext i |
| exact ⟨M,by rw [hM] |
|
|
| lemma triple_pairing_grid {s l d : ℕ} (hs : 2*l*d ≤ s) (k : Fin 3 → Fin (d+1)) : |
| ∃ M : Pairing (TripleVertices s), crossVector tripleColor M = |
| (fun a => (![1,0,0] a : ℝ) + (2*l:ℝ)*(k a:ℝ)) := by |
| let E : Fin 6 → ℕ := ![s-l*((k 0:ℕ)+k 1), 1+2*l*k 0, 2*l*k 1, |
| s-l*((k 0:ℕ)+k 2), 2*l*k 2, s-l*((k 1:ℕ)+k 2)] |
| have hk (a : Fin 3) : (k a : ℕ) ≤ d := Nat.le_of_lt_succ (k a).isLt |
| have hle (a b : Fin 3) : l*((k a:ℕ)+k b) ≤ s := by |
| calc |
| _ ≤ l*(d+d) := Nat.mul_le_mul_left l (Nat.add_le_add (hk a) (hk b)) |
| _ = 2*l*d := by ring |
| _ ≤ s := hs |
| have he : endpointSizes E = ![2*s+1,2*s+1,2*s] := by |
| ext i |
| fin_cases i < |
| · have h := Nat.sub_add_cancel (hle 0 1) |
| · have h := Nat.sub_add_cancel (hle 0 2) |
| · have h := Nat.sub_add_cancel (hle 1 2) |
| obtain ⟨M,hM⟩ := exists_pairing_of_endpoints tripleColor E (by rw [tripleColor_histogram, he]) |
| refine ⟨M,?_⟩ |
| funext i |
| rw [crossVector, hM] |
| fin_cases i < |
|
|
| end MatchingSmoothing |
|
|
| end OAI |
|
|
