openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /LeftDecoderKnownRows.lean
| import OAI.Computability.PerfectCompleteness.Decoding.LeftDecoder | |
| namespace OAI | |
| namespace PerfectCompleteness.LeftDecoderKnownRows | |
| open scoped Classical | |
| open UniqueGamesTheorem.Foundations.Games | |
| open ClauseSupport MixedSupport | |
| open LeftDecoder (F2) | |
| noncomputable section | |
| variable {n : Nat} {K Z : Type*} [AddCommGroup K] [Module F2 K] | |
| (slots : Fin n → Slot) (H : Submodule F2 (Assignment slots → F2)) | |
| (other : Assignment slots → Z) (labeling : KeyStrategy.Strategy n) | |
| (W : Submodule F2 H) (s : (H ⧸ W) →ₗ[F2] H) | |
| (hs : W.mkQ.comp s = LinearMap.id) | |
| theorem exists_fallback_candidate_realizer | |
| (hW : RepresentativeMatrixTable.Determined slots H other W) : | |
| ∃ x : Assignment slots, | |
| other x = (CanonicalMatrixTable.response slots H other labeling | |
| (0 : CanonicalMatrixTable.Matrix (K := K) slots H)).2 ∧ | |
| ∀ q : PulledColumnSpace.Candidates W ⊥ | |
| (RepresentativeMatrixTable.correctionFunctional slots H other W s hs | |
| (CanonicalMatrixTable.response slots H other labeling | |
| (0 : CanonicalMatrixTable.Matrix (K := K) slots H)).2), | |
| ∀ b : W, q.val (b : H) = b.val.val x := by | |
| obtain ⟨x, _, hx⟩ := CanonicalMatrixTable.response_realized slots H other labeling | |
| (0 : CanonicalMatrixTable.Matrix (K := K) slots H) | |
| refine ⟨x, hx, ?_⟩ | |
| intro q b | |
| rw [PulledColumnSpace.candidate_apply_known, ← hx] | |
| simpa only [add_zero] using RepresentativeDecoder.coset_apply_at_image | |
| slots H other W s hs hW x 0 W.dualAnnihilator.zero_mem b | |
| attribute [local instance] UniqueGamesTheorem.Appendix.RankLevelFilter.linearMapFintype | |
| variable [FiniteDimensional F2 H] | |
| [Fintype H] [Fintype K] | |
| (useful : (Module.Dual F2 (H ⧸ W) →ₗ[F2] K) → Prop) | |
| local instance representativeDualFintype : Fintype (Module.Dual F2 H) := | |
| LeftDecoder.dualFintype (V := H) | |
| theorem adviceLaw_probability_eq_one_of_known_realizer | |
| (hW : RepresentativeMatrixTable.Determined slots H other W) | |
| (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) (A : ManyGoodRows.RowMap K r) | |
| (U : Module.Dual F2 (H ⧸ W) →ₗ[F2] (Fin r → F2)) | |
| (event : Module.Dual F2 H → Bool) | |
| (hevent : ∀ q : Module.Dual F2 H, | |
| (∃ x : Assignment slots, ∀ b : W, q (b : H) = b.val.val x) → event q = true) : | |
| (LeftDecoder.adviceLaw slots H other labeling W s hs useful r ρ A U).probability | |
| event = 1 := by | |
| by_cases hgood : ManyGoodRows.GoodRow | |
| (RepresentativeMatrixTable.partialTable slots H other labeling W s useful) r ρ A U | |
| · rw [LeftDecoder.adviceLaw_of_good slots H other labeling W s hs useful r ρ A U hgood] | |
| unfold LeftDecoder.law | |
| apply LeftDecoder.affineLaw_probability_eq_one | |
| intro q | |
| obtain ⟨x, _, hknown⟩ := LeftDecoder.exists_candidate_realizer | |
| slots H other labeling W s hs useful hW | |
| (ManyGoodRows.selectWitness | |
| (RepresentativeMatrixTable.partialTable slots H other labeling W s useful) | |
| r ρ A U hgood) hρ | |
| exact hevent q.val ⟨x, hknown q⟩ | |
| · simp only [LeftDecoder.adviceLaw, dite_eq_right hgood] | |
| apply LeftDecoder.affineLaw_probability_eq_one | |
| intro q | |
| obtain ⟨x, _, hknown⟩ := | |
| exists_fallback_candidate_realizer (K := K) slots H other labeling W s hs hW | |
| exact hevent q.val ⟨x, hknown q⟩ | |
| theorem adviceLaw_known_realizer | |
| (hW : RepresentativeMatrixTable.Determined slots H other W) | |
| (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) (A : ManyGoodRows.RowMap K r) | |
| (U : Module.Dual F2 (H ⧸ W) →ₗ[F2] (Fin r → F2)) : | |
| (LeftDecoder.adviceLaw slots H other labeling W s hs useful r ρ A U).probability | |
| (fun q => decide (∃ x : Assignment slots, | |
| ∀ b : W, q (b : H) = b.val.val x)) = 1 := | |
| adviceLaw_probability_eq_one_of_known_realizer | |
| slots H other labeling W s hs useful hW r ρ hρ A U _ | |
| (fun _ hx => decide_eq_true hx) | |
| end | |
| end PerfectCompleteness.LeftDecoderKnownRows | |
| end OAI | |