| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Extraction.InitialHash |
| import OAI.LinearAlgebra.MatrixFields.Histories.InitialOrbit |
|
|
| namespace OAI |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldInitialGibbs |
|
|
| open AllFieldParameters AllFieldHistory AllFieldInitialEntropy AllFieldInitialOrbit |
| open MatrixMultiplication.Foundation |
| open scoped BigOperators |
|
|
| def coordinateWeight (k : Fin 17) : ℝ := |
| (weight (initialInteger k.val) : ℝ) |
|
|
| def orderedWeight (u : JointPopulation.Shape) : ℝ := |
| ∏ s : Fin 3, coordinateWeight (JointPopulation.shapeSide s u) |
|
|
| def initialNormalizer : ℝ := ((sortedInitial.map initialWeight).sum : ℚ) |
|
|
| def initialPotential (k : Fin 17) : ℝ := Real.log (coordinateWeight k) |
|
|
| def initialLogNormalizer : ℝ := Real.log initialNormalizer |
|
|
| theorem coordinateWeight_pos (k : Fin 17) : 0 < coordinateWeight k := by |
| apply Rat.cast_pos.mpr |
| apply weight_positive_of_argument_bound |
| exact assigned_argument_bound _ tablea initial_table_arguments_bounded _ |
|
|
| theorem orderedWeight_pos (u : JointPopulation.Shape) : 0 < orderedWeight u := |
| Finset.prod_pos (fun _ _ => coordinateWeight_pos _) |
|
|
| theorem initialNormalizer_pos : 0 < initialNormalizer := by |
| apply Rat.cast_pos.mpr |
| exact list_sum_map_positive _ _ (by decide +kernel) |
| (fun s _ => initialWeight_positive s) |
|
|
| theorem orderedWeight_source (p : PlacedInitial) : |
| orderedWeight (initialShapeCode p) = |
| ∏ s : Fin 3, (weight (initialInteger (sortedInitial.get p.1 s)) : ℝ) := by |
| simp only [orderedWeight, coordinateWeight, initialShapeCode_side] |
| exact p.2.symm.prod_comp |
| (fun s : Fin 3 => (weight (initialInteger (sortedInitial.get p.1 s)) : ℝ)) |
|
|
| theorem initialWeight_source (p : PlacedInitial) : |
| (initialWeight (sortedInitial.get p.1) : ℝ) = |
| (initialMultiplicity (sortedInitial.get p.1) : ℝ) * |
| orderedWeight (initialShapeCode p) := by |
| rw [orderedWeight_source] |
| simp only [initialWeight] |
| push_cast |
| rfl |
|
|
| theorem placedInitialLaw_mass (p : PlacedInitial) : |
| placedInitialLaw.mass p = |
| (initialMultiplicity (sortedInitial.get p.1) : ℝ) * |
| orderedWeight (initialShapeCode p) / initialNormalizer / 6 := by |
| change (initialLaw (sortedInitial.get p.1) : ℝ) / 6 = _ |
| rw [initialLaw, ite_eq_left (List.get_mem _ _), Rat.cast_div, initialWeight_source p] |
| rfl |
|
|
| theorem orderedInitialLaw_mass (u : JointPopulation.Shape) |
| (hu : u.1.val + u.2.1.val + u.2.2.val = 16) : |
| orderedInitialLaw.mass u = orderedWeight u / initialNormalizer := by |
| classical |
| have hcount : (∑ p : PlacedInitial, |
| if initialShapeCode p = u then |
| (initialMultiplicity (sortedInitial.get p.1) : ℝ) else 0) = 6 := by |
| exact_mod_cast weighted_fiber_count u hu |
| simp only [orderedInitialLaw, FiniteLaw.map_mass, placedInitialLaw_mass] |
| calc |
| _ = (∑ p : PlacedInitial, if initialShapeCode p = u then |
| (initialMultiplicity (sortedInitial.get p.1) : ℝ) else 0) * |
| (orderedWeight u / initialNormalizer / 6) := by |
| rw [Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro p _ |
| by_cases hp : initialShapeCode p = u |
| · simp only [hp, ite_true] |
| ring |
| · simp only [hp, ite_false, zero_mul] |
| _ = _ := by rw [hcount]; ring |
|
|
| theorem orderedInitialLaw_zero (u : JointPopulation.Shape) |
| (hu : u.1.val + u.2.1.val + u.2.2.val ≠ 16) : |
| orderedInitialLaw.mass u = 0 := by |
| classical |
| simp only [orderedInitialLaw, FiniteLaw.map_mass] |
| apply Finset.sum_eq_zero |
| intro p _ |
| apply ite_eq_right |
| intro hp |
| have ht := source_total p |
| rw [hp] at ht |
| exact hu ht |
|
|
| theorem orderedInitialLaw_mass_eq (u : JointPopulation.Shape) : |
| orderedInitialLaw.mass u = |
| if u.1.val + u.2.1.val + u.2.2.val = 16 |
| then orderedWeight u / initialNormalizer else 0 := by |
| split_ifs with hu |
| · exact orderedInitialLaw_mass u hu |
| · exact orderedInitialLaw_zero u hu |
|
|
| theorem orderedInitialLaw_positive (u : JointPopulation.Shape) |
| (hu : u.1.val + u.2.1.val + u.2.2.val = 16) : |
| 0 < orderedInitialLaw.mass u := by |
| rw [orderedInitialLaw_mass u hu] |
| exact div_pos (orderedWeight_pos u) initialNormalizer_pos |
|
|
| theorem orderedInitialLaw_support (u : JointPopulation.Shape) |
| (hu : 0 < orderedInitialLaw.mass u) : |
| u.1.val + u.2.1.val + u.2.2.val = 16 := by |
| by_contra hn |
| rw [orderedInitialLaw_zero u hn] at hu |
| exact lt_irrefl _ hu |
|
|
| theorem log_orderedWeight (u : JointPopulation.Shape) : |
| Real.log (orderedWeight u) = |
| initialPotential (JointPopulation.shapeSide 0 u) + |
| initialPotential (JointPopulation.shapeSide 1 u) + |
| initialPotential (JointPopulation.shapeSide 2 u) := by |
| rw [orderedWeight, Real.log_prod |
| (fun s _ => ne_of_gt (coordinateWeight_pos (JointPopulation.shapeSide s u)))] |
| simp only [Fin.sum_univ_three, initialPotential] |
|
|
| theorem orderedInitialLaw_log (u : JointPopulation.Shape) |
| (hu : 0 < orderedInitialLaw.mass u) : |
| Real.log (orderedInitialLaw.mass u) = |
| initialPotential (JointPopulation.shapeSide 0 u) + |
| initialPotential (JointPopulation.shapeSide 1 u) + |
| initialPotential (JointPopulation.shapeSide 2 u) - initialLogNormalizer := by |
| rw [orderedInitialLaw_mass u (orderedInitialLaw_support u hu), |
| Real.log_div (ne_of_gt (orderedWeight_pos u)) (ne_of_gt initialNormalizer_pos), |
| log_orderedWeight] |
| rfl |
|
|
| end MatrixMultiplication.AllFieldInitialGibbs |
|
|
| namespace MatrixMultiplication.AllFieldInitialFamily |
|
|
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory AllFieldFiniteFamily |
| open AllFieldInitialEntropy JointOrdinaryPopulationSelection Filter |
| open scoped BigOperators Topology |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {K : ℕ} (allocation : Allocation) |
|
|
| def grid : ℕ := populationLength (K := K) allocation 1 |
|
|
| theorem grid_pos : 0 < grid (K := K) allocation := |
| AllFieldPopulationCounts.blockLength_pos (amount (K := K) allocation) (by decide) |
|
|
| theorem populationLength_eq_mul_grid (dilation : ℕ) : |
| populationLength (K := K) allocation dilation = dilation * grid (K := K) allocation := by |
| simp only [grid, populationLength, AllFieldPopulationCounts.blockLength, one_mul] |
|
|
| theorem initialJointCounts_dilation_ratio {dilation : ℕ} (hd : 0 < dilation) |
| (j : Fin K) (u : JointPopulation.Shape) : |
| (initialJointCounts allocation dilation j u : ℝ) / (dilation : ℝ) = |
| (grid (K := K) allocation : ℝ) * orderedInitialLaw.mass u := by |
| rw [initialJointCounts_cast, populationLength_eq_mul_grid, Nat.cast_mul, mul_assoc] |
| have hd' : (dilation : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.ne_of_gt hd) |
| exact mul_div_cancel_left₀ _ hd' |
|
|
| theorem initialJointCounts_dilation_rate (j : Fin K) (u : JointPopulation.Shape) : |
| Tendsto (fun dilation : ℕ => |
| (initialJointCounts allocation dilation j u : ℝ) / (dilation : ℝ)) atTop |
| (𝓝 ((grid (K := K) allocation : ℝ) * orderedInitialLaw.mass u)) := by |
| apply tendsto_const_nhds.congr' |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with dilation hd |
| exact (initialJointCounts_dilation_ratio allocation hd j u).symm |
| |
| theorem native_dilation_capacity : |
| entropyRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) |
| (fun _ => orderedInitialLaw) - |
| degreeRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) |
| (fun _ => orderedInitialLaw) = |
| (grid (K := K) allocation : ℝ) * ((K : ℝ) * nativeH0) := by |
| simp only [entropyRate, degreeRate, sideDegreeRate, orderedInitialLaw_side_entropy, |
| Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, max_self] |
| ring |
| |
| def hashRate (ε : ℝ) : ℝ := |
| selectionHashRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) |
| (fun _ => orderedInitialLaw) ((grid (K := K) allocation : ℝ) * ε) |
| |
| theorem selection_exponent (ε : ℝ) (dilation : ℕ) : |
| (dilation : ℝ) * |
| (entropyRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) |
| (fun _ => orderedInitialLaw) - |
| degreeRate (fun _ : Fin K => (grid (K := K) allocation : ℝ)) |
| (fun _ => orderedInitialLaw) - (grid (K := K) allocation : ℝ) * ε) = |
| (populationLength (K := K) allocation dilation : ℝ) * ((K : ℝ) * nativeH0 - ε) := by |
| rw [native_dilation_capacity, populationLength_eq_mul_grid, Nat.cast_mul] |
| ring |
| |
| abbrev Sample (ε : ℝ) (dilation : ℕ) := |
| JointCoarseHashing.Sample (JointPopulation.Position |
| (initialJointCounts (K := K) allocation dilation)) |
| (JointOrdinarySelection.hashLevel (hashRate (K := K) allocation ε) dilation) |
| |
| def Good (ε : ℝ) (dilation : ℕ) |
| (s : Sample (K := K) allocation ε dilation) |
| (e : JointPopulation.Target (initialJointCounts (K := K) allocation dilation)) : Prop := |
| JointOrdinarySelection.Good |
| (JointOrdinarySelection.hashLevel (hashRate (K := K) allocation ε) dilation) |
| (JointOrdinarySelection.hashSet (hashRate (K := K) allocation ε) dilation) |
| (JointPopulation.ambientSet (initialJointCounts allocation dilation) (fun _ => 16)) |
| (JointPopulation.triple (initialJointCounts allocation dilation) e) s |
| |
| def executionWithWidth (F : Type*) [Field F] (ε residentWidth : ℝ) (dilation : ℕ) |
| (s : Sample (K := K) allocation ε dilation) |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))) |
| (hgood : ∀ e ∈ G, Good allocation ε dilation s e) : |
| Execution (cwSource F K (populationLength (K := K) allocation dilation)) |
| (Tensor.directSum (fun _ : AllFieldInitialHash.SelectedTargets |
| (initialJointCounts (K := K) allocation dilation) G => |
| stateTensor F (K := K) allocation dilation residentWidth 0)) := |
| AllFieldInitialState.execution allocation dilation F residentWidth |
| (AllFieldInitialHash.selectedExecution (F := F) |
| (initialJointCounts allocation dilation) (initialJointCounts_sum allocation dilation) |
| (initialJointCounts_support allocation dilation) |
| (JointOrdinarySelection.hashLevel (hashRate (K := K) allocation ε) dilation) |
| (JointOrdinarySelection.hashSet (hashRate (K := K) allocation ε) dilation) |
| (JointOrdinarySelection.hashSet_AP (hashRate (K := K) allocation ε) dilation) |
| s G hgood) |
| |
| @[simp] theorem executionWithWidth_copies (F : Type*) [Field F] |
| (ε residentWidth : ℝ) (dilation : ℕ) |
| (s : Sample (K := K) allocation ε dilation) |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))) |
| (hgood : ∀ e ∈ G, Good allocation ε dilation s e) : |
| Fintype.card |
| (executionWithWidth allocation F ε residentWidth dilation s G hgood).Copies = 1 := rfl |
| |
| def execution (F : Type*) [Field F] (ε : ℝ) (dilation : ℕ) |
| (s : Sample (K := K) allocation ε dilation) |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))) |
| (hgood : ∀ e ∈ G, Good allocation ε dilation s e) : |
| Execution (cwSource F K (populationLength (K := K) allocation dilation)) |
| (Tensor.directSum (fun _ : AllFieldInitialHash.SelectedTargets |
| (initialJointCounts (K := K) allocation dilation) G => |
| stateTensor F (K := K) allocation dilation ε 0)) := |
| executionWithWidth allocation F ε ε dilation s G hgood |
| |
| @[simp] theorem execution_copies (F : Type*) [Field F] (ε : ℝ) (dilation : ℕ) |
| (s : Sample (K := K) allocation ε dilation) |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))) |
| (hgood : ∀ e ∈ G, Good allocation ε dilation s e) : |
| Fintype.card (execution allocation F ε dilation s G hgood).Copies = 1 := rfl |
| |
| theorem eventually_exists_execution_with_width (F : Type*) [Field F] |
| (residentWidth : ℝ) {ε : ℝ} (hε : 0 < ε) : |
| ∀ᶠ dilation in atTop, |
| ∃ (s : Sample (K := K) allocation ε dilation) |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))), |
| G.card = ⌊Real.exp ((populationLength (K := K) allocation dilation : ℝ) * |
| ((K : ℝ) * nativeH0 - ε))⌋₊ ∧ |
| (∀ e ∈ G, Good allocation ε dilation s e) ∧ |
| ∃ E : Execution (cwSource F K (populationLength (K := K) allocation dilation)) |
| (Tensor.directSum (fun _ : AllFieldInitialHash.SelectedTargets |
| (initialJointCounts (K := K) allocation dilation) G => |
| stateTensor F (K := K) allocation dilation residentWidth 0)), |
| Fintype.card E.Copies = 1 := by |
| have hslack : 0 < (grid (K := K) allocation : ℝ) * ε := |
| mul_pos (by exact_mod_cast grid_pos (K := K) allocation) hε |
| have hselection := JointOrdinaryPopulationSelection.eventually_exists_selection |
| (fun dilation => initialJointCounts (K := K) allocation dilation) (fun _ => 16) |
| (fun _ : Fin K => (grid (K := K) allocation : ℝ)) (fun _ => orderedInitialLaw) |
| (fun _ => AllFieldInitialGibbs.initialPotential) |
| (fun _ => AllFieldInitialGibbs.initialPotential) |
| (fun _ => AllFieldInitialGibbs.initialPotential) |
| (fun _ => AllFieldInitialGibbs.initialLogNormalizer) |
| (fun _ => Nat.cast_nonneg _) |
| (fun dilation => initialJointCounts_support allocation dilation) |
| (fun _ => AllFieldInitialGibbs.orderedInitialLaw_positive) |
| (fun _ => AllFieldInitialGibbs.orderedInitialLaw_log) |
| (initialJointCounts_dilation_rate allocation) hslack |
| filter_upwards [hselection] with dilation hd |
| obtain ⟨s, G, hcard, hgood, _hpairwise⟩ := hd |
| rw [selection_exponent allocation ε dilation] at hcard |
| exact ⟨s, G, hcard, hgood, |
| executionWithWidth allocation F ε residentWidth dilation s G hgood, |
| executionWithWidth_copies allocation F ε residentWidth dilation s G hgood⟩ |
| |
| theorem eventually_exists_execution (F : Type*) [Field F] {ε : ℝ} (hε : 0 < ε) : |
| ∀ᶠ dilation in atTop, |
| ∃ (s : Sample (K := K) allocation ε dilation) |
| (G : Finset (JointPopulation.Target (initialJointCounts (K := K) allocation dilation))), |
| G.card = ⌊Real.exp ((populationLength (K := K) allocation dilation : ℝ) * |
| ((K : ℝ) * nativeH0 - ε))⌋₊ ∧ |
| (∀ e ∈ G, Good allocation ε dilation s e) ∧ |
| ∃ E : Execution (cwSource F K (populationLength (K := K) allocation dilation)) |
| (Tensor.directSum (fun _ : AllFieldInitialHash.SelectedTargets |
| (initialJointCounts (K := K) allocation dilation) G => |
| stateTensor F (K := K) allocation dilation ε 0)), |
| Fintype.card E.Copies = 1 := |
| eventually_exists_execution_with_width allocation F ε hε |
| |
| end MatrixMultiplication.AllFieldInitialFamily |
| |
| end |
| |
| end MatrixAllFields |
| |
| end OAI |
| |
