openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Machines /BinaryRankOneCount.lean
| import Mathlib.Algebra.Field.ZMod | |
| import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | |
| import Mathlib.LinearAlgebra.Matrix.Rank | |
| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayTransportLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.BinaryRankOneCount | |
| open scoped BigOperators Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| noncomputable section | |
| theorem rank_le_one_iff_vecMulVec {K I J : Type*} [Field K] | |
| [Fintype I] [Fintype J] (A : Matrix I J K) : | |
| A.rank ≤ 1 ↔ ∃ u : I → K, ∃ v : J → K, Matrix.vecMulVec u v = A := by | |
| constructor | |
| · intro hA | |
| let S : Submodule K (I → K) := Submodule.span K (Set.range A.col) | |
| have hS : Module.finrank K S ≤ 1 := by | |
| simpa only [S, Matrix.rank_eq_finrank_span_cols] using hA | |
| obtain ⟨u, hu⟩ := (finrank_le_one_iff (K := K) (V := S)).mp hS | |
| have hcol (j : J) : A.col j ∈ S := | |
| Submodule.subset_span (Set.mem_range_self j) | |
| choose v hv using (fun j : J => hu ⟨A.col j, hcol j⟩) | |
| refine ⟨u.val, v, ?_⟩ | |
| ext i j | |
| change u.val i * v j = A i j | |
| have hij := congrArg (fun z : S => z.val i) (hv j) | |
| change v j * u.val i = A i j at hij | |
| exact (mul_comm _ _).trans hij | |
| · rintro ⟨u, v, rfl⟩ | |
| exact Matrix.rank_vecMulVec_le u v | |
| theorem rank_le_one_card_le_vec_pairs {K I J : Type*} [Field K] | |
| [Fintype K] [Fintype I] [Fintype J] : | |
| Fintype.card {A : Matrix I J K // A.rank ≤ 1} ≤ | |
| Fintype.card ((I → K) × (J → K)) := by | |
| let outer : ((I → K) × (J → K)) → {A : Matrix I J K // A.rank ≤ 1} := | |
| fun p => ⟨Matrix.vecMulVec p.1 p.2, Matrix.rank_vecMulVec_le p.1 p.2⟩ | |
| apply Fintype.card_le_of_surjective outer | |
| intro A | |
| obtain ⟨u, v, huv⟩ := (rank_le_one_iff_vecMulVec A.val).mp A.property | |
| exact ⟨(u, v), Subtype.ext huv⟩ | |
| theorem binary_rank_le_one_card_le (m : Nat) : | |
| Fintype.card {A : Matrix (Fin m) (Fin m) (ZMod 2) // A.rank ≤ 1} ≤ | |
| 2 ^ (2 * m) := by | |
| let outer : ((Fin m → ZMod 2) × (Fin m → ZMod 2)) → | |
| {A : Matrix (Fin m) (Fin m) (ZMod 2) // A.rank ≤ 1} := | |
| fun p => ⟨Matrix.vecMulVec p.1 p.2, Matrix.rank_vecMulVec_le p.1 p.2⟩ | |
| have covers : Function.Surjective outer := by | |
| intro A | |
| obtain ⟨u, v, huv⟩ := (rank_le_one_iff_vecMulVec A.val).mp A.property | |
| exact ⟨(u, v), Subtype.ext huv⟩ | |
| have vector_card : Fintype.card (Fin m → ZMod 2) = 2 ^ m := by | |
| rw [Fintype.card_fun, ZMod.card, Fintype.card_fin] | |
| calc | |
| _ ≤ Fintype.card ((Fin m → ZMod 2) × (Fin m → ZMod 2)) := | |
| Fintype.card_le_of_surjective outer covers | |
| _ = 2 ^ m * 2 ^ m := by rw [Fintype.card_prod, vector_card] | |
| _ = 2 ^ (2 * m) := by rw [← pow_add, two_mul] | |
| theorem binary_matrix_card (m : Nat) : | |
| Fintype.card (Matrix (Fin m) (Fin m) (ZMod 2)) = 2 ^ (m * m) := by | |
| change Fintype.card (Fin m → Fin m → ZMod 2) = _ | |
| simp only [Fintype.card_fun, Fintype.card_fin, ZMod.card, pow_mul] | |
| theorem uniform_rank_le_one_probability_le (m : Nat) : | |
| (FiniteDistribution.uniform (Matrix (Fin m) (Fin m) (ZMod 2))).probability | |
| (fun A => decide (A.rank ≤ 1)) ≤ | |
| (2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) := by | |
| have hcount : | |
| (Fintype.card {A : Matrix (Fin m) (Fin m) (ZMod 2) // A.rank ≤ 1} : ℝ) ≤ | |
| (2 : ℝ) ^ (2 * m) := by | |
| exact_mod_cast binary_rank_le_one_card_le m | |
| calc | |
| _ = (∑ A : Matrix (Fin m) (Fin m) (ZMod 2), | |
| if A.rank ≤ 1 then (1 : ℝ) else 0) / | |
| (Fintype.card (Matrix (Fin m) (Fin m) (ZMod 2)) : ℝ) := by | |
| change (∑ A : Matrix (Fin m) (Fin m) (ZMod 2), | |
| if decide (A.rank ≤ 1) then | |
| 1 / (Fintype.card (Matrix (Fin m) (Fin m) (ZMod 2)) : ℝ) else 0) = _ | |
| simp only [div_eq_mul_inv, Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro A _ | |
| by_cases hA : A.rank ≤ 1 <;> simp [hA] | |
| _ = (Fintype.card {A : Matrix (Fin m) (Fin m) (ZMod 2) // A.rank ≤ 1} : ℝ) / | |
| (Fintype.card (Matrix (Fin m) (Fin m) (ZMod 2)) : ℝ) := by | |
| simp only [Fintype.card_subtype, Finset.sum_boole] | |
| _ ≤ (2 : ℝ) ^ (2 * m) / | |
| (Fintype.card (Matrix (Fin m) (Fin m) (ZMod 2)) : ℝ) := | |
| div_le_div_of_nonneg_right hcount (Nat.cast_nonneg _) | |
| _ = (2 : ℝ) ^ (2 * m) / (2 : ℝ) ^ (m * m) := by | |
| rw [binary_matrix_card] | |
| norm_cast | |
| end | |
| end PerfectCompleteness.BinaryRankOneCount | |
| end | |
| end OAI | |