| import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.ComplexCW75LabelHierarchy |
| import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexFiniteMMRealization |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ConditionalLabels |
|
|
| namespace OAI |
|
|
| /-! Finite coefficient tensors and their algebraic transformations. -/ |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
| open Filter |
|
|
| namespace MatrixMultiplication.ComplexWitness |
|
|
| open MatrixMultiplication.Foundation |
|
|
| structure FiniteRectangularWitness where |
| outer : ℕ |
| inner : ℕ |
| multiplicity : ℕ |
| rank : ℕ |
| outer_pos : 0 < outer |
| inner_pos : 0 < inner |
| multiplicity_pos : 0 < multiplicity |
| rank_pos : 0 < rank |
| order : ℕ |
| degree : ℕ |
| approximation : Tensor.PolynomialApproximation |
| (Tensor.directSum (fun _ : Fin multiplicity => |
| Tensor.matrixCoefficients (K := ℂ) (Fin outer) (Fin inner) (Fin outer))) |
| rank order degree |
|
|
| abbrev CurveCoordinate := CW75Primitive.Word |
| abbrev CurveLeaf := CW75LabelHierarchy.Scalar |
|
|
| def curveApproximation : |
| Tensor.PolynomialApproximation CW75Primitive.tensor 27 31 92 := |
| CW75Primitive.approximation |
|
|
| theorem curve_leaf_count : Fintype.card CurveLeaf = 75 := |
| CW75LabelHierarchy.scalar_card |
|
|
| theorem curve_leaf_coefficient (s : CurveLeaf) : |
| CW75Primitive.tensor (CW75LabelHierarchy.x s) (CW75LabelHierarchy.y s) |
| (CW75LabelHierarchy.z s) = 1 := |
| CW75LabelHierarchy.coefficient_one s |
|
|
| theorem curve_support_exhausted (x y z : CurveCoordinate) |
| (h : CW75Primitive.tensor x y z ≠ 0) : |
| ∃ s : CurveLeaf, |
| (CW75LabelHierarchy.x s, CW75LabelHierarchy.y s, CW75LabelHierarchy.z s) = |
| (x, y, z) := |
| CW75LabelHierarchy.support_exhausted x y z h |
|
|
| def curveFiber (u : CurveCoordinate) : Finset (List Elementary.BoundaryTerm) := |
| CW75LabelHierarchy.retainedWords.filter fun w => |
| CW75Primitive.listedX w 0 = u 0 ∧ |
| CW75Primitive.listedX w 1 = u 1 ∧ |
| CW75Primitive.listedX w 2 = u 2 |
|
|
| def curveFiberSize (u : CurveCoordinate) : ℕ := (curveFiber u).card |
|
|
| theorem mem_curveFiber (u : CurveCoordinate) (w : List Elementary.BoundaryTerm) : |
| w ∈ curveFiber u ↔ |
| w ∈ CW75LabelHierarchy.retainedWords ∧ CW75Primitive.listedX w = u := by |
| simp only [curveFiber, Finset.mem_filter] |
| constructor |
| · rintro ⟨hw, h0, h1, h2⟩ |
| refine ⟨hw, ?_⟩ |
| funext i |
| fin_cases i <;> assumption |
| · rintro ⟨hw, h⟩ |
| exact ⟨hw, congrFun h 0, congrFun h 1, congrFun h 2⟩ |
|
|
| def curveFiberEquiv (u : CurveCoordinate) : |
| {s : CurveLeaf // CW75LabelHierarchy.x s = u} ≃ ↥(curveFiber u) where |
| toFun s := ⟨s.val.val, (mem_curveFiber u s.val.val).2 |
| ⟨s.val.property, s.property⟩⟩ |
| invFun s := ⟨⟨s.val, ((mem_curveFiber u s.val).1 s.property).1⟩, |
| ((mem_curveFiber u s.val).1 s.property).2⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
|
|
| theorem curveFiberSize_eq_card (u : CurveCoordinate) : |
| curveFiberSize u = |
| Fintype.card {s : CurveLeaf // CW75LabelHierarchy.x s = u} := by |
| rw [Fintype.card_congr (curveFiberEquiv u), Fintype.card_coe] |
| rfl |
|
|
| theorem curveFiberSize_pos : ∀ u : CurveCoordinate, 0 < curveFiberSize u := by |
| decide |
|
|
| theorem curve_fiber_histogram : |
| (Finset.univ.filter fun u : CurveCoordinate => curveFiberSize u = 1).card = 8 ∧ |
| (Finset.univ.filter fun u : CurveCoordinate => curveFiberSize u = 2).card = 12 ∧ |
| (Finset.univ.filter fun u : CurveCoordinate => curveFiberSize u = 4).card = 6 ∧ |
| (Finset.univ.filter fun u : CurveCoordinate => curveFiberSize u = 19).card = 1 := by |
| decide |
|
|
| theorem curve_fiber_values : ∀ u : CurveCoordinate, |
| curveFiberSize u = 1 ∨ curveFiberSize u = 2 ∨ |
| curveFiberSize u = 4 ∨ curveFiberSize u = 19 := by |
| decide |
|
|
| theorem sum_curveLeaves_by_X (f : CurveCoordinate → ℝ) : |
| (∑ s : CurveLeaf, f (CW75LabelHierarchy.x s)) = |
| ∑ u : CurveCoordinate, (curveFiberSize u : ℝ) * f u := by |
| classical |
| rw [← Fintype.sum_fiberwise' CW75LabelHierarchy.x f] |
| simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, ← curveFiberSize_eq_card] |
|
|
| theorem sum_curveFiberSizes (f : ℕ → ℝ) : |
| (∑ u : CurveCoordinate, f (curveFiberSize u)) = |
| 8 * f 1 + 12 * f 2 + 6 * f 4 + f 19 := by |
| classical |
| rcases curve_fiber_histogram with ⟨h1, h2, h4, h19⟩ |
| have he (u : CurveCoordinate) : f (curveFiberSize u) = |
| (if curveFiberSize u = 1 then f 1 else 0) + |
| (if curveFiberSize u = 2 then f 2 else 0) + |
| (if curveFiberSize u = 4 then f 4 else 0) + |
| (if curveFiberSize u = 19 then f 19 else 0) := by |
| rcases curve_fiber_values u with h | h | h | h <;> simp [h] |
| simp_rw [he] |
| simp only [Finset.sum_add_distrib, ← Finset.sum_filter, Finset.sum_const, |
| nsmul_eq_mul, h1, h2, h4, h19, Nat.cast_ofNat, Nat.cast_one, one_mul] |
|
|
| abbrev CurveLabel := CW75LabelHierarchy.Label |
|
|
| def curveLabels : (n : ℕ) → CurveLeaf → CurveLabel n |
| | 0 => CW75LabelHierarchy.outer |
| | 1 => CW75LabelHierarchy.bPositions |
| | 2 => CW75LabelHierarchy.bChoices |
| | 3 => CW75LabelHierarchy.cPositions |
| | 4 => CW75LabelHierarchy.aChoices |
| | _ + 5 => fun _ _ => false |
|
|
| def curveFirstSide : ℕ → CW75LabelHierarchy.Side |
| | 3 | 4 => .y |
| | _ => .x |
|
|
| def curveSecondSide : ℕ → CW75LabelHierarchy.Side |
| | 1 | 3 | 4 => .z |
| | _ => .y |
|
|
| def curveFirstReader : (n : ℕ) → LabelRecord CurveLabel n → CurveCoordinate → CurveLabel n |
| | 0, _ => CW75LabelHierarchy.outerX |
| | 1, _ => CW75LabelHierarchy.bPositionsX |
| | 2, _ => CW75LabelHierarchy.bChoicesX |
| | 3, _ => CW75LabelHierarchy.cPositionsY |
| | 4, prior => CW75LabelHierarchy.aChoicesY prior.1.1.2 prior.2 |
| | _ + 5, _ => fun _ _ => false |
|
|
| def curveSecondReader : (n : ℕ) → LabelRecord CurveLabel n → CurveCoordinate → CurveLabel n |
| | 0, _ => CW75LabelHierarchy.outerY |
| | 1, prior => CW75LabelHierarchy.bPositionsZ prior.2 |
| | 2, prior => CW75LabelHierarchy.bChoicesY prior.2 |
| | 3, prior => CW75LabelHierarchy.cPositionsZ prior.1.1.2 |
| | 4, _ => CW75LabelHierarchy.aChoicesZ |
| | _ + 5, _ => fun _ _ => false |
|
|
| theorem curveFirstReader_readable (n : ℕ) (hn : n < 5) (s : CurveLeaf) : |
| curveFirstReader n (labelRecordOf curveLabels n s) |
| (CW75LabelHierarchy.originalCoordinate (curveFirstSide n) s) = |
| curveLabels n s := by |
| interval_cases n |
| · exact CW75LabelHierarchy.outerX_read s |
| · exact CW75LabelHierarchy.bPositionsX_read s |
| · exact CW75LabelHierarchy.bChoicesX_read s |
| · exact CW75LabelHierarchy.cPositionsY_read s |
| · exact CW75LabelHierarchy.aChoicesY_read s |
|
|
| theorem curveSecondReader_readable (n : ℕ) (hn : n < 5) (s : CurveLeaf) : |
| curveSecondReader n (labelRecordOf curveLabels n s) |
| (CW75LabelHierarchy.originalCoordinate (curveSecondSide n) s) = |
| curveLabels n s := by |
| interval_cases n |
| · exact CW75LabelHierarchy.outerY_read s |
| · exact CW75LabelHierarchy.bPositionsZ_read s |
| · exact CW75LabelHierarchy.bChoicesY_read s |
| · exact CW75LabelHierarchy.cPositionsZ_read s |
| · exact CW75LabelHierarchy.aChoicesZ_read s |
|
|
| theorem curve_completeRecord_injective : |
| Function.Injective (labelRecordOf curveLabels 5) := by |
| intro a b h |
| apply CW75LabelHierarchy.positionLabels_injective |
| exact Prod.ext (congrArg (fun q : LabelRecord CurveLabel 5 => q.1.1.1.2) h) |
| (Prod.ext (congrArg (fun q : LabelRecord CurveLabel 5 => q.1.2) h) |
| (Prod.ext (congrArg (fun q : LabelRecord CurveLabel 5 => q.2) h) |
| (congrArg (fun q : LabelRecord CurveLabel 5 => q.1.1.2) h))) |
|
|
| def encodeCurveX (u : CurveCoordinate) : LabelRecord CurveLabel 3 := |
| (((PUnit.unit, CW75LabelHierarchy.outerX u), |
| CW75LabelHierarchy.bPositionsX u), CW75LabelHierarchy.bChoicesX u) |
|
|
| def decodeCurveX (r : LabelRecord CurveLabel 3) : CurveCoordinate := fun i => |
| if r.2 i then 1 else if r.1.2 i then 2 else 0 |
|
|
| theorem decode_encodeCurveX (u : CurveCoordinate) : |
| decodeCurveX (encodeCurveX u) = u := by |
| funext i |
| simp only [decodeCurveX, encodeCurveX, CW75LabelHierarchy.bPositionsX, |
| CW75LabelHierarchy.bChoicesX] |
| have hd : ∀ v : Fin 3, |
| (if decide (v = 1) then (1 : Fin 3) else if decide (v ≠ 0) then 2 else 0) = v := by |
| decide |
| exact hd (u i) |
|
|
| theorem encodeCurveX_injective : Function.Injective encodeCurveX := |
| Function.LeftInverse.injective decode_encodeCurveX |
|
|
| theorem curve_incidentPrefix (s : CurveLeaf) : |
| labelRecordOf curveLabels 3 s = encodeCurveX (CW75LabelHierarchy.x s) := rfl |
|
|
| theorem curve_incidentPrefix_eq_iff (s t : CurveLeaf) : |
| labelRecordOf curveLabels 3 s = labelRecordOf curveLabels 3 t ↔ |
| CW75LabelHierarchy.x s = CW75LabelHierarchy.x t := by |
| rw [curve_incidentPrefix, curve_incidentPrefix, encodeCurveX_injective.eq_iff] |
|
|
| theorem curve_incidentPrefix_entropy (p : FiniteLaw CurveLeaf) : |
| finiteEntropy (p.map (labelRecordOf curveLabels 3)).mass = |
| finiteEntropy (p.map CW75LabelHierarchy.x).mass := by |
| classical |
| have hmass : (p.map (labelRecordOf curveLabels 3)).mass = |
| ((p.map CW75LabelHierarchy.x).map encodeCurveX).mass := by |
| funext r |
| by_cases hr : ∃ u, encodeCurveX u = r |
| · obtain ⟨u, rfl⟩ := hr |
| rw [FiniteLaw.map_mass_apply _ _ encodeCurveX_injective] |
| simp only [FiniteLaw.map_mass, curve_incidentPrefix, encodeCurveX_injective.eq_iff] |
| · have hn (u : CurveCoordinate) : encodeCurveX u ≠ r := fun h => hr ⟨u, h⟩ |
| simp [FiniteLaw.map_mass, curve_incidentPrefix, hn] |
| rw [hmass] |
| exact FiniteLaw.map_entropy_of_injective _ _ encodeCurveX_injective |
|
|
| def curveReaderPair (n : Fin 5) : ReaderPair := |
| match n.val with |
| | 1 => .xz |
| | 3 | 4 => .yz |
| | _ => .xy |
|
|
| theorem curveReaderPair_incident_prefix (n : Fin 5) : |
| (curveReaderPair n = .xy ∨ curveReaderPair n = .xz) ↔ n.val < 3 := by |
| fin_cases n <;> decide |
|
|
| theorem curveIncidentRate_eq_entropy_X (p : FiniteLaw CurveLeaf) : |
| ConditionalLabels.lawPairingRate p curveLabels 5 curveReaderPair .xy + |
| ConditionalLabels.lawPairingRate p curveLabels 5 curveReaderPair .xz = |
| finiteEntropy (p.map CW75LabelHierarchy.x).mass := by |
| rw [ConditionalLabels.incident_rate_eq_prefix_entropy p curveLabels 5 3 |
| (by decide) curveReaderPair curveReaderPair_incident_prefix] |
| exact curve_incidentPrefix_entropy p |
|
|
| theorem curveHiddenRate_eq_entropy_sub_X (p : FiniteLaw CurveLeaf) : |
| ConditionalLabels.lawPairingRate p curveLabels 5 curveReaderPair .yz = |
| finiteEntropy p.mass - finiteEntropy (p.map CW75LabelHierarchy.x).mass := by |
| rw [ConditionalLabels.remaining_rate_eq_entropy_sub_prefix p curveLabels 5 3 |
| (by decide) curveReaderPair curveReaderPair_incident_prefix curve_completeRecord_injective, |
| curve_incidentPrefix_entropy] |
|
|
| theorem exists_finite_score_gt |
| (W : ℕ → FiniteMMRealization) {β L : ℝ} |
| (hlimit : Tendsto (fun n => (W n).rates.score β) atTop (nhds L)) |
| (hgap : Real.log 3 < L) : |
| ∃ n, Real.log 3 < (W n).rates.score β := |
| (hlimit.eventually_const_lt hgap).exists |
|
|
| theorem omega_lt_of_score_limit |
| (W : ℕ → FiniteMMRealization) {β L : ℝ} |
| (hlimit : Tendsto (fun n => (W n).rates.score β) atTop (nhds L)) |
| (hgap : Real.log 3 < L) : Arithmetic.omega < β := by |
| obtain ⟨n, hn⟩ := exists_finite_score_gt W hlimit hgap |
| exact (W n).omega_lt_of_score_gt hn |
|
|
| end MatrixMultiplication.ComplexWitness |
|
|
| end |
|
|
| end OAI |
|
|
