openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /ScheduledPositivity.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ScheduledYield | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.InitialGibbs | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldScheduledPositivity | |
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory | |
| open AllFieldInitialEntropy AllFieldNativeCapacity AllFieldScheduledYield Filter | |
| open scoped BigOperators Topology | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| theorem nativeH0_pos : 0 < nativeH0 := by | |
| let p := orderedInitialLaw.map (JointPopulation.shapeSide 0) | |
| have hzero : 0 < p.mass 0 := by | |
| have hp := AllFieldInitialGibbs.orderedInitialLaw_positive | |
| ((0 : Fin 17), (0 : Fin 17), (16 : Fin 17)) (by decide) | |
| exact hp.trans_le (orderedInitialLaw.mass_le_map_mass | |
| (JointPopulation.shapeSide 0) ((0 : Fin 17), (0 : Fin 17), (16 : Fin 17))) | |
| have hone : 0 < p.mass 1 := by | |
| have hp := AllFieldInitialGibbs.orderedInitialLaw_positive | |
| ((1 : Fin 17), (0 : Fin 17), (15 : Fin 17)) (by decide) | |
| exact hp.trans_le (orderedInitialLaw.mass_le_map_mass | |
| (JointPopulation.shapeSide 0) ((1 : Fin 17), (0 : Fin 17), (15 : Fin 17))) | |
| have hpair : p.mass 0 + p.mass 1 ≤ 1 := by | |
| have hs : (∑ k ∈ ({0, 1} : Finset (Fin 17)), p.mass k) ≤ | |
| ∑ k : Fin 17, p.mass k := | |
| Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) | |
| (fun k _ _ => p.nonneg k) | |
| simpa [p.total] using hs | |
| have hterm : 0 < entropyTerm (p.mass 0) := | |
| mul_pos_of_neg_of_neg (neg_neg_of_pos hzero) (Real.log_neg hzero (by linarith)) | |
| have hentropy : 0 < finiteEntropy p.mass := hterm.trans_le | |
| (Finset.single_le_sum | |
| (fun k _ => entropyTerm_nonneg (p.nonneg k) (p.mass_le_one k)) | |
| (Finset.mem_univ 0)) | |
| rw [← orderedInitialLaw_side_entropy 0] | |
| have hm : p.mass = JointPopulationRates.sideMass orderedInitialLaw.mass 0 := by | |
| funext k | |
| simp only [p, FiniteLaw.map_mass, JointPopulationRates.sideMass] | |
| rw [← hm] | |
| exact hentropy | |
| theorem initial_capacity_pos (K : ℕ) (hK : 0 < K) : | |
| 0 < (K : ℝ) * nativeH0 := mul_pos (Nat.cast_pos.mpr hK) nativeH0_pos | |
| theorem tickCapacity_pos (K tick : ℕ) (hK : 2 ≤ K) (htick : tick < K + 2) | |
| (a b c : Staggering.Capacity) | |
| (ha : ∀ i, 0 < a i) (hb : ∀ i, 0 < b i) (hc : ∀ i, 0 < c i) | |
| (i : Fin 3) : 0 < FiniteSchedule.tickCapacity K a b c tick i := by | |
| have hA := ha i | |
| have hB := hb i | |
| have hC := hc i | |
| unfold FiniteSchedule.tickCapacity | |
| split_ifs <;> first | |
| | positivity | |
| | (simp only [FiniteSchedule.stageActive, Fin.val_zero, Fin.val_one] at *; omega) | |
| theorem physical_order_capacity_pos (K tick : ℕ) (hK : 2 ≤ K) | |
| (htick : tick < K + 2) (allocation : Allocation) (sigma : Placement) | |
| (ha : ∀ i, 0 < nativeLA i) (hb : ∀ i, 0 < nativeLB i) | |
| (hc : ∀ i, 0 < nativeLC (fun j => (allocation.mass j : ℝ)) i) : | |
| 0 < Staggering.minCapacity | |
| (physicalTickCapacity allocation tick (workCapacity (K := K)) sigma) := by | |
| rw [physical_order_native_capacity, PhysicalOrders.minCapacity_mul _ _ (by norm_num)] | |
| exact mul_pos (by norm_num) (Staggering.minCapacity_pos _ | |
| (tickCapacity_pos K tick hK htick _ _ _ ha hb hc)) | |
| theorem eventually_physical_order_capacity_pos (K : ℕ) (hK : 2 ≤ K) | |
| (allocation : ℕ → Allocation) | |
| (h : ∀ i, Tendsto (fun j => ((allocation j).mass i : ℝ)) atTop (𝓝 (nativeLambda i))) | |
| (ha : ∀ i, 0 < nativeLA i) (hb : ∀ i, 0 < nativeLB i) | |
| (hc : ∀ i, 0 < nativeLC nativeLambda i) : | |
| ∀ᶠ j in atTop, ∀ tick < K + 2, ∀ sigma : Placement, | |
| 0 < Staggering.minCapacity | |
| (physicalTickCapacity (allocation j) tick (workCapacity (K := K)) sigma) := by | |
| have he : ∀ᶠ j in atTop, ∀ i, | |
| 0 < nativeLC (fun k => ((allocation j).mass k : ℝ)) i := | |
| Filter.eventually_all.mpr fun i => (nativeLC_tendsto h i).eventually_const_lt (hc i) | |
| filter_upwards [he] with j hj | |
| exact fun tick htick sigma => | |
| physical_order_capacity_pos K tick hK htick (allocation j) sigma ha hb hj | |
| end MatrixMultiplication.AllFieldScheduledPositivity | |
| end | |
| end OAI | |