openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Counting /CompanionCanonicalEncoding.lean
| import OAI.Combinatorics.ContingencyTables.Machines.CanonicalInitialEncoding | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionInitialDraw | |
| import OAI.Combinatorics.ContingencyTables.Counting.CompanionOrderedProposal | |
| namespace OAI | |
| /-! # A common explicit encoding for initialization and proposals | |
| Cells follow `cellEnum`, their copies follow increasing finite indices, | |
| and each copy contributes its false and true coordinates consecutively. | |
| The polynomial initializer computes exactly this encoding of the actual | |
| initial draw. The same order may therefore be used by the proposal scan. | |
| -/ | |
| namespace ContingencyTables.CompanionCanonicalEncoding | |
| open CommonBasesFPRAS TwoLiftStates CompanionFactorialApplication | |
| open InitialTransversals InitialFactorialBits CompanionInitialDraw | |
| open CompanionArithmeticProposal CompanionOrderedProposal | |
| open scoped BigOperators Classical | |
| noncomputable section | |
| variable {I J : Type} [Fintype I] [Fintype J] | |
| def copyEquiv (A : Instance I J) : Labels A ≃ Copy (cellCapacity A) := | |
| Equiv.sigmaCongrLeft' (cellEnum A) | |
| def pairEquiv (A : Instance I J) : (Labels A×Bool) ≃ (Copy (cellCapacity A)×Bool) := | |
| Equiv.prodCongr (copyEquiv A) (Equiv.refl Bool) | |
| def coordinateOrder (A : Instance I J) : (Labels A×Bool) ≃ | |
| Fin (CanonicalInitialEncoding.pairOrder (cellCapacity A)).length := | |
| (pairEquiv A).trans (CanonicalInitialEncoding.coordinateEquiv (cellCapacity A)) | |
| lemma encodeSet_reindex (A : Instance I J) (S : Set (Labels A×Bool)) : | |
| encodeSet (coordinateOrder A) S= | |
| encodeSet (CanonicalInitialEncoding.coordinateEquiv (cellCapacity A)) | |
| ((pairEquiv A).symm ⁻¹' S) := rfl | |
| /-- Reindexing the physical initial draw gives the independent cell | |
| choices used by the actual list initializer. -/ | |
| lemma initial_preimage (A : Instance I J) (B : ℕ) | |
| (bits : Fin (initialBits A B) → Bool) : | |
| (pairEquiv A).symm ⁻¹' (draw A B bits).val.val= | |
| encode (cellCapacity A) (drawChoices (cellCapacity A) B bits) := by | |
| ext p | |
| rcases p with ⟨⟨i,k⟩,side⟩ | |
| have hc := Equiv.piCongrLeft'_symm_apply_apply | |
| (fun s : SmallCell A => Finset (Fin (A.b s.val.1 s.val.2))) | |
| (cellEnum A) (drawChoices (cellCapacity A) B bits) i | |
| cases side | |
| · change k∈(choiceEquiv (fun s : SmallCell A => A.b s.val.1 s.val.2) | |
| (cellEnum A) (drawChoices (cellCapacity A) B bits)) ((cellEnum A).symm i) ↔ | |
| k∈drawChoices (cellCapacity A) B bits i | |
| exact Iff.of_eq (congrArg (fun S => k∈S) hc) | |
| · change k∉(choiceEquiv (fun s : SmallCell A => A.b s.val.1 s.val.2) | |
| (cellEnum A) (drawChoices (cellCapacity A) B bits)) ((cellEnum A).symm i) ↔ | |
| k∉drawChoices (cellCapacity A) B bits i | |
| exact Iff.of_eq (congrArg (fun S => k∉S) hc) | |
| /-- The polynomial binary initializer emits precisely the code of the | |
| literal `PlainIndex` output with its distributional error bound. -/ | |
| theorem initialCode_eq (A : Instance I J) (B : ℕ) (hB : 0<B) | |
| (bits : Fin (initialBits A B) → Bool) : | |
| CanonicalInitialEncoding.initialCode ((List.ofFn (cellCapacity A),B),List.ofFn bits)= | |
| encodeSet (coordinateOrder A) (draw A B bits).val.val := by | |
| rw [encodeSet_reindex,initial_preimage] | |
| exact CanonicalInitialEncoding.initialCode_ofFn (cellCapacity A) B hB bits | |
| /-- The canonical initializer encoding can be fed directly to an | |
| arithmetic proposal with the same dyadic transition law. -/ | |
| theorem canonicalProposal_law (A : Instance I J) (S : CompanionBooleanPhase.State A) : | |
| FirstSuccess.mapLaw | |
| (FirstSuccess.uniformLaw (α := Fin (CompanionBooleanPhase.proposalBits A) → Bool)) | |
| (orderedProposal A (coordinateOrder A) S)= | |
| FirstSuccess.mapLaw | |
| (FirstSuccess.uniformLaw (α := Fin (CompanionBooleanPhase.proposalBits A) → Bool)) | |
| (CompanionBooleanPhase.proposal A S) := | |
| orderedProposal_law A (coordinateOrder A) S | |
| end | |
| end ContingencyTables.CompanionCanonicalEncoding | |
| end OAI | |