File size: 3,070 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Support
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryBranchWeights
open AllFieldHistory AllFieldHistorySupport AllFieldParameters
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
theorem stageB_branchPopulation_rat {K : ℕ} (allocation : Allocation) (dilation : ℕ)
(h : APositive K) (b : BSplit h) (phi : Placement) :
(branchPopulation allocation dilation (.stageB h, phi) b : ℚ) =
(population allocation dilation (.afterA h.val, phi) : ℚ) *
stageBLaw (aShape h.val) (bSplit ⟨h, b, false⟩) := by
simp only [branchPopulation, Work.child, population_cast, amount, canonicalAmount, bAmount]
ring
theorem stageB_branchPopulation_real {K : ℕ} (allocation : Allocation) (dilation : ℕ)
(h : APositive K) (b : BSplit h) (phi : Placement) :
(branchPopulation allocation dilation (.stageB h, phi) b : ℝ) =
(population allocation dilation (.afterA h.val, phi) : ℝ) *
(stageBLaw (aShape h.val) (bSplit ⟨h, b, false⟩) : ℝ) := by
exact_mod_cast stageB_branchPopulation_rat allocation dilation h b phi
theorem stageB_branchWeight_rat {K dilation : ℕ} (allocation : Allocation)
(hd : 0 < dilation) (h : APositive K) (b : BSplit h) (phi : Placement) :
(branchPopulation allocation dilation (.stageB h, phi) b : ℚ) /
(population allocation dilation (.afterA h.val, phi) : ℚ) =
stageBLaw (aShape h.val) (bSplit ⟨h, b, false⟩) := by
have hpos := work_source_population_pos allocation hd (.stageB h, phi)
have hden : (population allocation dilation (.afterA h.val, phi) : ℚ) ≠ 0 := by
exact_mod_cast Nat.ne_of_gt hpos
apply (div_eq_iff hden).2
rw [stageB_branchPopulation_rat]
exact mul_comm _ _
theorem stageB_branchWeight_real {K dilation : ℕ} (allocation : Allocation)
(hd : 0 < dilation) (h : APositive K) (b : BSplit h) (phi : Placement) :
(branchPopulation allocation dilation (.stageB h, phi) b : ℝ) /
(population allocation dilation (.afterA h.val, phi) : ℝ) =
(stageBLaw (aShape h.val) (bSplit ⟨h, b, false⟩) : ℝ) := by
have hpos := work_source_population_pos allocation hd (.stageB h, phi)
have hden : (population allocation dilation (.afterA h.val, phi) : ℝ) ≠ 0 := by
exact_mod_cast Nat.ne_of_gt hpos
apply (div_eq_iff hden).2
rw [stageB_branchPopulation_real]
exact mul_comm _ _
theorem shapeCounts_weighted_sum {I : Type*} [Fintype I]
(code : I → JointPopulation.Shape) (weight : I → ℕ)
(f : JointPopulation.Shape → ℝ) :
(∑ u, (shapeCounts code weight u : ℝ) * f u) =
∑ b, (weight b : ℝ) * f (code b) := by
simp only [shapeCounts, Nat.cast_sum, Nat.cast_ite, Nat.cast_zero, Finset.sum_mul]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro b _
simp [ite_mul]
end MatrixMultiplication.AllFieldHistoryBranchWeights
end
end OAI
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