openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldHistory /BranchWeights.lean
| 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 | |