File size: 3,868 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.NativeLaws
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupBranchLaws
open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory
open AllFieldHistoryChildLaws AllFieldGroupOrbitData AllFieldActiveLaws
open AllFieldGroupNativeLaws
attribute [local instance] Classical.propDecidable Classical.decEq
theorem jointCounts_branch_pos {K dilation : ℕ} (allocation : Allocation)
(hd : 0 < dilation) (w : PlacedWork K) (b : w.1.Branch)
(hb : 0 < (branchLaw w.1).mass b) :
0 < jointCounts allocation dilation w (branchShape w b) := by
have hp : 0 < (population allocation dilation (w.1.source, w.2) : ℝ) := by
exact_mod_cast work_source_population_pos allocation hd w
have hc : 0 < (jointCounts allocation dilation w (branchShape w b) : ℝ) := by
rw [jointCounts_cast, placedLaw_mass_branch]
exact mul_pos hp hb
exact_mod_cast hc
theorem branchLaw_stageA_mass_pos {K : ℕ} (h : InitialPositive K)
(b : (Work.stageA h).Branch) :
0 < (branchLaw (Work.stageA h)).mass b := by
change 0 < (stageALaw (initialShape h.val) (aSplit ⟨h, b, false⟩) : ℝ)
exact_mod_cast stageALaw_positive _ _ (aSplit_mem ⟨h, b, false⟩)
theorem branchLaw_stageB_mass_pos {K : ℕ} (h : APositive K)
(b : (Work.stageB h).Branch) :
0 < (branchLaw (Work.stageB h)).mass b := by
change 0 < (stageBLaw (aShape h.val) (bSplit ⟨h, b, false⟩) : ℝ)
exact_mod_cast stageBLaw_positive _ _ (bSplit_mem ⟨h, b, false⟩)
theorem supportedHalfLaw_branchShape_of_mass_pos {K tick : ℕ} {sigma : Placement}
(allocation : Allocation) (right : Bool) (h : ActiveOrder K tick sigma)
(b : h.val.val.1.Branch) (hb : 0 < (branchLaw h.val.val.1).mass b)
(side : Fin 3) :
supportedHalfLaw allocation right h.val (branchShape h.val.val b) (sigma side) =
fun a => (h.val.val.1.childLaw
(halfShape h.val.val.1.parentShape (h.val.val.1.splitShape b) right)
(h.val.val.1.priority side) a : ℝ) := by
have hc : 0 < activeCounts allocation 1 h.val (branchShape h.val.val b) :=
jointCounts_branch_pos allocation (by decide) h.val.val b hb
rw [supportedHalfLaw_eq_of_counts_pos allocation 1 right h.val _ hc]
funext a
rw [halfLawAt_eq, canonicalChildShape_branchShape, placement_symm_order_side]
theorem supportedHalfLaw_branchShape_of_mass_ne_zero {K tick : ℕ}
{sigma : Placement} (allocation : Allocation) (right : Bool)
(h : ActiveOrder K tick sigma) (b : h.val.val.1.Branch)
(hb : (branchLaw h.val.val.1).mass b ≠ 0) (side : Fin 3) :
supportedHalfLaw allocation right h.val (branchShape h.val.val b) (sigma side) =
fun a => (h.val.val.1.childLaw
(halfShape h.val.val.1.parentShape (h.val.val.1.splitShape b) right)
(h.val.val.1.priority side) a : ℝ) := by
exact supportedHalfLaw_branchShape_of_mass_pos allocation right h b
(lt_of_le_of_ne ((branchLaw h.val.val.1).nonneg b) (Ne.symm hb)) side
theorem weighted_supportedHalfLaw_branchShape {K tick : ℕ} {sigma : Placement}
(allocation : Allocation) (right : Bool) (h : ActiveOrder K tick sigma)
(b : h.val.val.1.Branch) (side : Fin 3) (a : Statistic h.val) :
(branchLaw h.val.val.1).mass b *
supportedHalfLaw allocation right h.val (branchShape h.val.val b) (sigma side) a =
(branchLaw h.val.val.1).mass b *
(h.val.val.1.childLaw
(halfShape h.val.val.1.parentShape (h.val.val.1.splitShape b) right)
(h.val.val.1.priority side) a : ℝ) := by
by_cases hb : (branchLaw h.val.val.1).mass b = 0
· simp only [hb, zero_mul]
· rw [supportedHalfLaw_branchShape_of_mass_ne_zero allocation right h b hb side]
end MatrixMultiplication.AllFieldGroupBranchLaws
end
end OAI
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