openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /InitialHistories.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Counts | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldHistory | |
| open AllFieldParameters | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| theorem shapeCounts_fiber {I : Type*} [Fintype I] (code : I → JointPopulation.Shape) | |
| (weight : I → ℕ) (u : JointPopulation.Shape) : | |
| shapeCounts code weight u = ∑ i : {i : I // code i = u}, weight i.val := by | |
| unfold shapeCounts | |
| rw [← Finset.sum_filter] | |
| exact Finset.sum_subtype _ (by intro i; simp) weight | |
| def shapeCountFiberEquiv {I : Type*} [Fintype I] (code : I → JointPopulation.Shape) | |
| (weight : I → ℕ) (u : JointPopulation.Shape) : | |
| Fin (shapeCounts code weight u) ≃ | |
| (Σ i : {i : I // code i = u}, Fin (weight i.val)) := | |
| Fintype.equivOfCardEq (by | |
| simp only [Fintype.card_fin, Fintype.card_sigma, shapeCounts_fiber]) | |
| abbrev InitialPlacement (K : ℕ) := Initial K × Placement | |
| abbrev InitialShapeFiber {K : ℕ} (j : Fin K) (u : JointPopulation.Shape) := | |
| {p : Fin sortedInitial.length × Placement // rootShape (j, p.1) p.2 = u} | |
| def initialClassPositionEquiv {K : ℕ} (allocation : Allocation) (dilation : ℕ) | |
| (j : Fin K) (u : JointPopulation.Shape) : | |
| Fin (initialJointCounts allocation dilation j u) ≃ | |
| (Σ p : InitialShapeFiber j u, | |
| Fin (population allocation dilation (.initial (j, p.val.1), p.val.2))) := | |
| shapeCountFiberEquiv _ _ u | |
| theorem initialClassPosition_shape {K : ℕ} (allocation : Allocation) (dilation : ℕ) | |
| (j : Fin K) (u : JointPopulation.Shape) | |
| (i : Fin (initialJointCounts allocation dilation j u)) : | |
| rootShape (j, (initialClassPositionEquiv allocation dilation j u i).1.val.1) | |
| (initialClassPositionEquiv allocation dilation j u i).1.val.2 = u := | |
| (initialClassPositionEquiv allocation dilation j u i).1.property | |
| def initialStateEquiv (K : ℕ) : State K 0 ≃ InitialPlacement K where | |
| toFun h := by | |
| rcases h with ⟨⟨h, phi⟩, hr⟩ | |
| cases h with | |
| | initial g => exact (g, phi) | |
| | afterA a => have := hr.1; dsimp [resident] at hr; omega | |
| | afterB b => have := hr.1; dsimp [resident] at hr; omega | |
| | partC c => dsimp [resident] at hr; omega | |
| | afterC c => dsimp [resident] at hr; omega | |
| invFun h := ⟨(.initial h.1, h.2), fun _ => Nat.zero_le _⟩ | |
| left_inv h := by | |
| rcases h with ⟨⟨h, phi⟩, hr⟩ | |
| cases h with | |
| | initial g => rfl | |
| | afterA a => have := hr.1; dsimp [resident] at hr; omega | |
| | afterB b => have := hr.1; dsimp [resident] at hr; omega | |
| | partC c => dsimp [resident] at hr; omega | |
| | afterC c => dsimp [resident] at hr; omega | |
| right_inv h := rfl | |
| end MatrixMultiplication.AllFieldHistory | |
| end | |
| end OAI | |