| import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.NativeLaws |
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|
| namespace OAI |
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|
| /-! Group assignments, orbit counts and extraction capacities. -/ |
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|
| noncomputable section |
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|
| namespace MatrixMultiplication.AllFieldGroupBranchLaws |
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|
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory |
| open AllFieldHistoryChildLaws AllFieldGroupOrbitData AllFieldActiveLaws |
| open AllFieldGroupNativeLaws |
| attribute [local instance] Classical.propDecidable Classical.decEq |
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|
| 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 |
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| end |
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| end OAI |
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|
