| import OAI.Computability.PerfectCompleteness.Foundations.ManyGoodRowsLemmas |
| import OAI.Computability.PerfectCompleteness.Foundations.RecursiveSpaceEquivLemmas |
| import OAI.Computability.PerfectCompleteness.Sampling.CutSamplerReplayTransportLemmas |
|
|
| namespace OAI |
|
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| section |
|
|
| namespace PerfectCompleteness.GoodRowSampling |
|
|
| noncomputable section |
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|
| open scoped BigOperators Classical |
| open UniqueGamesTheorem.Fourier.MatrixCharacters (F2) |
| open UniqueGamesTheorem.Foundations.Games |
| open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype) |
| open PerfectCompleteness.ManyGoodRows |
|
|
| attribute [local instance] linearMapFintype |
|
|
| variable {E F Y : Type*} |
| [AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F] |
| [FiniteDimensional F2 E] [FiniteDimensional F2 F] |
| [Fintype E] [Fintype F] [Fintype Y] |
|
|
| abbrev RowFiber {r : Nat} (A : RowMap F r) (U : E →ₗ[F2] (Fin r → F2)) := |
| {X : E →ₗ[F2] F // A.comp X = U} |
|
|
| def rowFiberEquiv {r : Nat} (A : RowMap F r) (U : E →ₗ[F2] (Fin r → F2)) |
| (T : E →ₗ[F2] F) (hT : A.comp T = U) : |
| MatrixSlice.Slice (⊥ : Submodule F2 E) (LinearMap.ker A) T ≃ RowFiber A U where |
| toFun X := ⟨X.val, (RowAdvice.slice_row_eq ⊥ A T X).trans hT⟩ |
| invFun X := ⟨X.val, by |
| apply LinearMap.ext |
| intro x |
| have hx : (x : E) = 0 := (Submodule.mem_bot F2).mp x.property |
| simp [hx], (RowAdvice.comp_eq_iff_ker_quotient A X.val T).mp (X.property.trans hT.symm)⟩ |
| left_inv X := by apply Subtype.ext; rfl |
| right_inv X := by apply Subtype.ext; rfl |
|
|
| omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype E] [Fintype F] in |
| @[simp] theorem rowFiberEquiv_val {r : Nat} (A : RowMap F r) |
| (U : E →ₗ[F2] (Fin r → F2)) (T : E →ₗ[F2] F) (hT : A.comp T = U) |
| (X : MatrixSlice.Slice (⊥ : Submodule F2 E) (LinearMap.ker A) T) : |
| (rowFiberEquiv A U T hT X).val = X.val := rfl |
|
|
| omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype Y] in |
| theorem witness_rowFiber_nonempty {f : (E →ₗ[F2] F) → Option Y} |
| {r : Nat} {ρ : ℝ} {A : RowMap F r} {U : E →ₗ[F2] (Fin r → F2)} |
| (w : Witness f r ρ A U) : Nonempty (RowFiber A U) := |
| ⟨⟨w.base, w.row_value⟩⟩ |
|
|
| omit [Fintype Y] in |
| theorem witness_column_probability {f : (E →ₗ[F2] F) → Option Y} |
| {r : Nat} {ρ : ℝ} {A : RowMap F r} {U : E →ₗ[F2] (Fin r → F2)} |
| (w : Witness f r ρ A U) : |
| letI : Nonempty (RowFiber A U) := witness_rowFiber_nonempty w |
| 1 / (2 : ℝ) ^ (r * Module.finrank F2 F) ≤ |
| (FiniteDistribution.uniform (RowFiber A U)).probability |
| (fun X => decide (X.val.domRestrict w.columnSpace = w.base.domRestrict w.columnSpace)) := by |
| let : Nonempty (RowFiber A U) := witness_rowFiber_nonempty w |
| have h := SliceMass.probability_columnConstraint_ge_of_finrank_le |
| w.columnSpace (LinearMap.ker A) w.base r w.column_count |
| have he := UniformConditioning.uniform_probability_equiv |
| (rowFiberEquiv A U w.base w.row_value) |
| (fun X : RowFiber A U => |
| decide (X.val.domRestrict w.columnSpace = w.base.domRestrict w.columnSpace)) |
| apply h.trans_eq |
| refine Eq.trans ?_ he |
| congr 1 |
| funext X |
| exact decide_eq_decide.mpr Iff.rfl |
|
|
| omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype Y] in |
| theorem witness_row_probability_pos {f : (E →ₗ[F2] F) → Option Y} |
| {r : Nat} {ρ : ℝ} {A : RowMap F r} {U : E →ₗ[F2] (Fin r → F2)} |
| (w : Witness f r ρ A U) : |
| 0 < (FiniteDistribution.uniform (E →ₗ[F2] F)).probability |
| (fun X => decide (A.comp X = U)) := by |
| let : Nonempty (RowFiber A U) := witness_rowFiber_nonempty w |
| apply lt_of_lt_of_eq |
| (UniformConditioning.uniform_probability_pos (fun X : E →ₗ[F2] F => A.comp X = U)) |
| congr 1 |
| funext X |
| exact decide_eq_decide.mpr Iff.rfl |
|
|
| omit [Fintype Y] in |
| theorem witness_conditional_probability {f : (E →ₗ[F2] F) → Option Y} |
| {r : Nat} {ρ : ℝ} {A : RowMap F r} {U : E →ₗ[F2] (Fin r → F2)} |
| (w : Witness f r ρ A U) : |
| 1 / (2 : ℝ) ^ (r * Module.finrank F2 F) ≤ |
| ((FiniteDistribution.uniform (E →ₗ[F2] F)).condition |
| (fun X => decide (A.comp X = U)) (witness_row_probability_pos w)).probability |
| (fun X => decide (X.domRestrict w.columnSpace = w.base.domRestrict w.columnSpace)) := by |
| let : Nonempty (RowFiber A U) := witness_rowFiber_nonempty w |
| have hevent : (fun X : E →ₗ[F2] F => decide (A.comp X = U)) = |
| (fun X : E →ₗ[F2] F => @decide (A.comp X = U) (Classical.propDecidable _)) := by |
| funext X |
| exact decide_eq_decide.mpr Iff.rfl |
| simp only [hevent, UniformConditioning.probability_condition_uniform] |
| convert witness_column_probability w using 1 |
| congr! |
|
|
| end |
| end PerfectCompleteness.GoodRowSampling |
|
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| end |
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| end OAI |
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|
