| import Mathlib |
| import OAI.Computability.MinUncut.PCP.GraphTables |
|
|
| namespace OAI |
|
|
| section |
| namespace MinUncutGames.Foundations.PCP.PortTableLookup |
|
|
| theorem flatMap_word {α β : Type*} (f : α → List β) |
| (width j : Nat) (hj : j < width) : |
| ∀ (rows : List α), (∀ a ∈ rows, (f a).length = width) → |
| ∀ (i : Nat) (row : α), rows[i]? = some row → |
| (rows.flatMap f)[width * i + j]? = (f row)[j]? := by |
| intro rows |
| induction rows with |
| | nil => |
| intro _ i row selected |
| simp at selected |
| | cons a rows ih => |
| intro hwidth i row selected |
| have ha : (f a).length = width := hwidth a (by simp) |
| cases i with |
| | zero => |
| have heq : a = row := by simpa using selected |
| subst row |
| simp only [Nat.mul_zero, Nat.zero_add, List.flatMap_cons] |
| exact List.getElem?_append_left (by omega) |
| | succ i => |
| have selected' : rows[i]? = some row := by simpa using selected |
| have hskip : (f a).length ≤ width * (i + 1) + j := by |
| rw [ha, Nat.mul_add, Nat.mul_one] |
| omega |
| have hoff : width * (i + 1) + j - (f a).length = width * i + j := by |
| rw [ha, Nat.mul_add, Nat.mul_one] |
| omega |
| rw [List.flatMap_cons, List.getElem?_append_right hskip, hoff] |
| exact ih (fun b hb => hwidth b (by simp [hb])) i row selected' |
|
|
| variable {n d : Nat} |
|
|
| theorem vertices_word (table : PortTables.Table n d) : |
| (PortTables.tableWords table)[0]? = some n := rfl |
|
|
| theorem darts_word (table : PortTables.Table n d) : |
| (PortTables.tableWords table)[1]? = some (n * d) := rfl |
|
|
| private theorem row_word (table : PortTables.Table n d) (i : Fin (n * d)) |
| (j : Nat) (hj : j < 4098) : |
| (PortTables.tableWords table)[2 + 4098 * i.val + j]? = |
| (GraphTables.rowWords (PortTables.flatRows table)[i])[j]? := by |
| rw [PortTables.tableWords_eq] |
| change (n :: n * d :: (PortTables.flatRows table).toList.flatMap |
| GraphTables.rowWords)[2 + 4098 * i.val + j]? = _ |
| rw [show 2 + 4098 * i.val + j = (4098 * i.val + j + 1) + 1 by omega, |
| List.getElem?_cons_succ, List.getElem?_cons_succ] |
| apply flatMap_word GraphTables.rowWords 4098 j hj |
| (PortTables.flatRows table).toList |
| (fun row _ => GraphTables.rowWords_length row) i.val |
| ((PortTables.flatRows table)[i]) |
| simp only [Vector.getElem?_toList, Vector.getElem?_eq_getElem i.isLt, |
| Fin.getElem_fin] |
|
|
| theorem tail_word (table : PortTables.Table n d) (i : Fin (n * d)) : |
| (PortTables.tableWords table)[2 + 4098 * i.val]? = |
| some (((PortTables.rowIndex n d).symm i).1.val) := by |
| have h := row_word table i 0 (by decide) |
| simpa only [Nat.add_zero, GraphTables.rowWords, PortTables.flatRows, |
| Fin.getElem_fin, Vector.getElem_ofFn, Fin.eta, List.cons_append, |
| List.nil_append, List.getElem?_cons_zero] using h |
|
|
| theorem reverse_word (table : PortTables.Table n d) (i : Fin (n * d)) : |
| (PortTables.tableWords table)[3 + 4098 * i.val]? = |
| some (table.reverseIndex[i].val) := by |
| have h := row_word table i 1 (by decide) |
| rw [show 2 + 4098 * i.val + 1 = 3 + 4098 * i.val by omega] at h |
| simpa only [GraphTables.rowWords, PortTables.flatRows, Fin.getElem_fin, |
| Vector.getElem_ofFn, Fin.eta, List.cons_append, List.nil_append, |
| List.getElem?_cons_succ, List.getElem?_cons_zero] using h |
|
|
| theorem relation_word (table : PortTables.Table n d) (i : Fin (n * d)) |
| (j : Fin 4096) : |
| (PortTables.tableWords table)[4 + 4098 * i.val + j.val]? = |
| some (GraphTables.bitWord table.relations[i][j]) := by |
| have h := row_word table i (j.val + 2) (by omega) |
| rw [show 2 + 4098 * i.val + (j.val + 2) = 4 + 4098 * i.val + j.val by omega] at h |
| simpa only [GraphTables.rowWords, GraphTables.relationWords, |
| PortTables.flatRows, Fin.getElem_fin, Vector.getElem_ofFn, Fin.eta, |
| List.cons_append, List.nil_append, List.getElem?_cons_succ, |
| List.getElem?_map, Vector.getElem?_toList, |
| Vector.getElem?_eq_getElem j.isLt, Option.map_some] using h |
|
|
| theorem accepts_word (table : PortTables.Table n d) (e : Fin n × Fin d) |
| (a b : PortTables.Label) : |
| (PortTables.tableWords table)[4 + 4098 * (PortTables.rowIndex n d e).val + |
| (GraphTables.relationIndex (a, b)).val]? = |
| some (GraphTables.bitWord (PortTables.accepts table e a b)) := |
| relation_word table (PortTables.rowIndex n d e) (GraphTables.relationIndex (a, b)) |
|
|
| theorem rotation_head_word (table : PortTables.Table n d) (e : Fin n × Fin d) : |
| (PortTables.tableWords table)[2 + 4098 * (table.reverseIndex[PortTables.rowIndex n d e]).val]? = |
| some ((PortTables.rotation table e).1.val) := |
| tail_word table (table.reverseIndex[PortTables.rowIndex n d e]) |
|
|
| theorem rotor_lookups (table : PortTables.Table n d) (e : Fin n × Fin d) : |
| (PortTables.tableWords table)[3 + 4098 * (PortTables.rowIndex n d e).val]? = |
| some (table.reverseIndex[PortTables.rowIndex n d e].val) ∧ |
| (PortTables.tableWords table)[2 + 4098 * (table.reverseIndex[PortTables.rowIndex n d e]).val]? = |
| some ((PortTables.rotation table e).1.val) := |
| ⟨reverse_word table (PortTables.rowIndex n d e), rotation_head_word table e⟩ |
|
|
| end MinUncutGames.Foundations.PCP.PortTableLookup |
|
|
| end |
|
|
| end OAI |
|
|
