openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /ActiveCapacity.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ScheduledYield | |
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.NativeLaws | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldActiveCapacity | |
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory | |
| open AllFieldNativeCapacity AllFieldActiveLaws AllFieldScheduledYield | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| def canonicalSideMass {K : ℕ} (w : Work K) (s : Fin 3) (k : Fin 17) : ℝ := | |
| ∑ b : w.Branch, if w.splitShape b s = k.val then (branchLaw w).mass b else 0 | |
| theorem placedLaw_priority_sideMass {K : ℕ} (w : PlacedWork K) | |
| (i : Fin 3) : | |
| JointPopulationRates.sideMass (placedLaw w).mass (w.physicalOrder i) = | |
| canonicalSideMass w.1 (w.1.priority i) := by | |
| funext k | |
| rw [placedLaw_sideMass] | |
| simp only [Fin.ext_iff, branchShape, encodePhysicalShape_side, | |
| PlacedWork.physicalOrder, Equiv.trans_apply, Equiv.symm_apply_apply, | |
| canonicalSideMass, branchLaw_mass] | |
| theorem canonicalSideMass_stageA {K : ℕ} (h : InitialPositive K) (s : Fin 3) : | |
| canonicalSideMass (.stageA h) s = | |
| (stageAMarginalLaw (initialShape h.val) h.property s).mass := by | |
| funext k | |
| rw [show (stageAMarginalLaw (initialShape h.val) h.property s).mass k = | |
| marginal (below (initialShape h.val)) | |
| (fun u => (stageALaw (initialShape h.val) u : ℝ)) s k from | |
| indexedMarginalLaw_mass _ _ _ _ _ _ k] | |
| change (∑ b : ASplit h, if (below (initialShape h.val))[b.val] s = k.val | |
| then (stageALaw (initialShape h.val) (below (initialShape h.val))[b.val] : ℝ) | |
| else 0) = _ | |
| exact Fin.sum_univ_fun_getElem _ | |
| (fun u : Shape => if u s = k.val then (stageALaw (initialShape h.val) u : ℝ) else 0) | |
| theorem canonicalSideMass_stageB {K : ℕ} (h : APositive K) (s : Fin 3) : | |
| canonicalSideMass (.stageB h) s = | |
| (stageBMarginalLaw (aShape h.val) h.property s).mass := by | |
| funext k | |
| rw [show (stageBMarginalLaw (aShape h.val) h.property s).mass k = | |
| marginal (below (aShape h.val)) | |
| (fun u => (stageBLaw (aShape h.val) u : ℝ)) s k from | |
| indexedMarginalLaw_mass _ _ _ _ _ _ k] | |
| change (∑ b : BSplit h, if (below (aShape h.val))[b.val] s = k.val | |
| then (stageBLaw (aShape h.val) (below (aShape h.val))[b.val] : ℝ) | |
| else 0) = _ | |
| exact Fin.sum_univ_fun_getElem _ | |
| (fun u : Shape => if u s = k.val then (stageBLaw (aShape h.val) u : ℝ) else 0) | |
| theorem canonicalSideMass_stageC {K : ℕ} (h : PartC K) (s : Fin 3) : | |
| canonicalSideMass (.stageC h) s = | |
| AllFieldStageCCapacity.stageCMarginal (cParameterParent h) (cShapeParent h) s := rfl | |
| theorem workCapacity_first {K : ℕ} (w : PlacedWork K) : | |
| workCapacity w.1 0 = finiteEntropy | |
| (JointPopulationRates.sideMass (placedLaw w).mass (w.physicalOrder 0)) := by | |
| rw [placedLaw_priority_sideMass] | |
| rcases w with ⟨w, phi⟩ | |
| cases w with | |
| | stageA h => | |
| rw [canonicalSideMass_stageA] | |
| exact stageACapacity_first_entropy _ h.property | |
| | stageB h => | |
| rw [canonicalSideMass_stageB] | |
| exact stageBCapacity_first_entropy _ h.property | |
| | stageC h => | |
| rw [canonicalSideMass_stageC] | |
| exact workCapacity_stageC h 0 | |
| theorem workCapacity_stageC_physical {K : ℕ} (h : PartC K) | |
| (phi : Placement) (i : Fin 3) : | |
| workCapacity (.stageC h) i = finiteEntropy | |
| (JointPopulationRates.sideMass (placedLaw (.stageC h, phi)).mass | |
| ((PlacedWork.physicalOrder (.stageC h, phi)) i)) := by | |
| rw [placedLaw_priority_sideMass, canonicalSideMass_stageC] | |
| exact workCapacity_stageC h i | |
| theorem order_first_capacity {K tick : ℕ} {sigma : Placement} | |
| (h : ActiveOrder K tick sigma) : | |
| workCapacity h.val.val.1 0 = finiteEntropy | |
| (JointPopulationRates.sideMass (orderLaw h).mass (sigma 0)) := by | |
| have hh := workCapacity_first h.val.val | |
| rw [h.property] at hh | |
| exact hh | |
| def orderEntropyRate {K tick : ℕ} (allocation : Allocation) (sigma : Placement) : ℝ := | |
| ∑ h : ActiveOrder K tick sigma, | |
| ((populationLength (K := K) allocation 1 : ℝ) * orderMass allocation h) * | |
| finiteEntropy (orderLaw h).mass | |
| def orderNativeDegree {K tick : ℕ} (allocation : Allocation) (sigma : Placement) | |
| (i : Fin 3) : ℝ := | |
| ∑ h : ActiveOrder K tick sigma, | |
| ((populationLength (K := K) allocation 1 : ℝ) * orderMass allocation h) * | |
| (finiteEntropy (orderLaw h).mass - workCapacity h.val.val.1 i) | |
| theorem order_entropy_sub_native_degree (K tick : ℕ) (allocation : Allocation) | |
| (sigma : Placement) : | |
| orderEntropyRate (K := K) (tick := tick) allocation sigma - | |
| max (orderNativeDegree (K := K) (tick := tick) allocation sigma 0) | |
| (max (orderNativeDegree (K := K) (tick := tick) allocation sigma 1) | |
| (orderNativeDegree (K := K) (tick := tick) allocation sigma 2)) = | |
| (populationLength (K := K) allocation 1 : ℝ) * | |
| ((1 / 6 : ℝ) * Staggering.minCapacity | |
| (FiniteSchedule.tickCapacity K nativeLA nativeLB | |
| (nativeLC (fun i => (allocation.mass i : ℝ))) tick)) := by | |
| unfold orderEntropyRate orderNativeDegree | |
| rw [JointCapacityAlgebra.weighted_joint_capacity] | |
| simp_rw [order_dilation_capacity, ← mul_assoc] | |
| exact PhysicalOrders.minCapacity_mul _ _ (by positivity) | |
| end MatrixMultiplication.AllFieldActiveCapacity | |
| end | |
| end OAI | |