File size: 5,436 Bytes
cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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
|