| import OAI.Computability.UniqueGames.Games.UniformTarget |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace UniqueGamesTheorem.Explicit.UniformRoundingTarget |
|
|
| open UniqueGamesTheorem.Foundations.Target |
| open UniqueGamesTheorem.MachineUniformRounding |
| open UniqueGamesTheorem.MachineUniformCopyOrder |
|
|
| /-- Apply the concrete largest-remainder list of occurrence IDs to the original |
| list. Equal records at different indices remain separate input occurrences. -/ |
| def rows {α : Type*} (C : Nat) (es : List α) : List α := |
| (Rounding.copies (Rounding.denominator es.length 1 C) es.length |
| (fun _ => 1)).map es.get |
|
|
| theorem rows_eq_occurrenceMajor {α : Type*} (C : Nat) (es : List α) |
| (h : 0 < es.length) : rows C es = occurrenceMajor C es := by |
| unfold rows |
| rw [copies_uniform es.length C h] |
| simp only [List.map_flatten, List.map_ofFn, Function.comp_def, List.map_replicate] |
| change (List.ofFn (fun i => List.replicate C (es.get i))).flatten = |
| (es.map (fun e => List.replicate C e)).flatten |
| congr 1 |
| calc |
| List.ofFn (fun i => List.replicate C (es.get i)) = |
| (List.ofFn es.get).map (fun e => List.replicate C e) := List.map_ofFn.symm |
| _ = es.map (fun e => List.replicate C e) := by rw [List.ofFn_get] |
|
|
| /-- Exact multiplicities and full permutation records agree with the machine's |
| copy-major target; only the enumeration order changes. -/ |
| theorem rows_perm_construct {q : Nat} (C : Nat) (hC : 0 < C) (g : Instance q) : |
| (rows C g.constraints).Perm (UniformTarget.construct C hC g).constraints := by |
| rw [rows_eq_occurrenceMajor C g.constraints g.constraintCount_positive] |
| exact occurrenceMajor_perm_copyMajor C g.constraints |
|
|
| theorem selected_count_construct {q : Nat} (C : Nat) (hC : 0 < C) (g : Instance q) |
| (a : Fin g.vertices → Fin q) : |
| countSatisfied a (rows C g.constraints) = |
| countSatisfied a (UniformTarget.construct C hC g).constraints := |
| Integration.InstanceEquivalences.countSatisfied_perm a (rows_perm_construct C hC g) |
|
|
| end UniqueGamesTheorem.Explicit.UniformRoundingTarget |
|
|
| end |
|
|
| end OAI |
|
|
