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2f887db | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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
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