| import Mathlib.Data.Fin.VecNotation |
| import Mathlib.Data.Fintype.CardEmbedding |
| import Mathlib.Tactic.FinCases |
| import Mathlib.Tactic.NormNum |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! The concrete clone sample space. Global distinctness is a stronger |
| conditioning rule than distinctness only at repeated original variables. |
| It retains the same 3/100 bound and makes the number of copies uniform. -/ |
|
|
| namespace UniqueGamesTheorem.Outer.Clone100Triples |
|
|
| abbrev Index := Fin 100 |
| abbrev Triple := Index × Index × Index |
|
|
| def good (t : Triple) : Prop := |
| t.1 ≠ t.2.1 ∧ t.1 ≠ t.2.2 ∧ t.2.1 ≠ t.2.2 |
|
|
| instance : DecidablePred good := fun _ => inferInstanceAs (Decidable (_ ∧ _ ∧ _)) |
|
|
| abbrev GoodTriple := {t : Triple // good t} |
|
|
| def toEmbedding (t : GoodTriple) : Fin 3 ↪ Index where |
| toFun := ![t.val.1, t.val.2.1, t.val.2.2] |
| inj' := by |
| have h01 := t.property.1 |
| have h02 := t.property.2.1 |
| have h12 := t.property.2.2 |
| intro i j h |
| fin_cases i <;> fin_cases j <;> simp_all |
|
|
| def fromEmbedding (f : Fin 3 ↪ Index) : GoodTriple := |
| ⟨(f 0, f 1, f 2), by |
| constructor |
| · exact fun h => (by decide : (0 : Fin 3) ≠ 1) (f.injective h) |
| constructor |
| · exact fun h => (by decide : (0 : Fin 3) ≠ 2) (f.injective h) |
| · exact fun h => (by decide : (1 : Fin 3) ≠ 2) (f.injective h)⟩ |
|
|
| def embeddingEquiv : GoodTriple ≃ (Fin 3 ↪ Index) where |
| toFun := toEmbedding |
| invFun := fromEmbedding |
| left_inv t := by |
| apply Subtype.ext |
| rfl |
| right_inv f := by |
| ext i |
| fin_cases i <;> rfl |
|
|
| theorem card_goodTriple : Fintype.card GoodTriple = 970200 := by |
| rw [Fintype.card_congr embeddingEquiv, Fintype.card_embedding_eq] |
| norm_num [Nat.descFactorial_succ] |
|
|
| theorem card_triple : Fintype.card Triple = 1000000 := by |
| norm_num [Fintype.card_prod, Fintype.card_fin] |
|
|
| instance : Nonempty GoodTriple := |
| ⟨⟨(0, 1, 2), by decide⟩⟩ |
|
|
| end UniqueGamesTheorem.Outer.Clone100Triples |
|
|
| end |
|
|
| end OAI |
|
|
