openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /PairConditionalEntropy.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.PairMarginals | |
| namespace OAI | |
| /-! Tensor extraction over arbitrary fields and its asymptotic rate. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldPairConditionalEntropy | |
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory | |
| open AllFieldNativeCapacity AllFieldPairMarginals | |
| open scoped BigOperators | |
| theorem entropy_sub_eq_conditional_sum | |
| {A B C : Type*} [Fintype A] [Fintype B] [Fintype C] | |
| (p : FiniteLaw A) (q : FiniteLaw B) (m : FiniteLaw C) | |
| (f : A → C) (g : B → C) (hp : p.map f = m) (hq : q.map g = m) : | |
| finiteEntropy p.mass - finiteEntropy q.mass = | |
| ∑ k, m.mass k * (finiteEntropy (p.conditional f k).mass - | |
| finiteEntropy (q.conditional g k).mass) := by | |
| rw [p.entropy_eq_map_add_conditional f, q.entropy_eq_map_add_conditional g, hp, hq] | |
| simp only [mul_sub, Finset.sum_sub_distrib] | |
| ring | |
| def stageASplitWeight (g : Shape) (hg : g ∈ positiveInitial) (side : Fin 3) : | |
| Fin (below g).length → Fin 17 := | |
| fun j => ⟨(below g).get j side, | |
| Nat.lt_succ_of_le ((below_le (List.get_mem _ _) side).trans | |
| ((positiveInitial_shape_spec g hg).1 side))⟩ | |
| def stageBSplitWeight (t : Shape) (ht : t ∈ positiveSecond) (side : Fin 3) : | |
| Fin (below t).length → Fin 17 := | |
| fun j => ⟨(below t).get j side, | |
| Nat.lt_succ_of_le ((below_le (List.get_mem _ _) side).trans | |
| ((positiveSecond_shape_spec t ht).1 side))⟩ | |
| theorem stageASplitLaw_ownWeight_marginal (g : Shape) (hg : g ∈ positiveInitial) | |
| (side : Fin 3) : | |
| (stageASplitLaw g hg).map (stageASplitWeight g hg side) = | |
| stageAMarginalLaw g hg side := rfl | |
| theorem stageBSplitLaw_ownWeight_marginal (t : Shape) (ht : t ∈ positiveSecond) | |
| (side : Fin 3) : | |
| (stageBSplitLaw t ht).map (stageBSplitWeight t ht side) = | |
| stageBMarginalLaw t ht side := rfl | |
| theorem stageA_entropy_sub_pair_eq_conditional_sum | |
| (g : Shape) (hg : g ∈ positiveInitial) (side : Fin 3) : | |
| finiteEntropy (stageASplitLaw g hg).mass - | |
| finiteEntropy (stageAPairLaw g hg side).mass = | |
| ∑ k, (stageAMarginalLaw g hg side).mass k * | |
| (finiteEntropy ((stageASplitLaw g hg).conditional | |
| (stageASplitWeight g hg side) k).mass - | |
| finiteEntropy ((stageAPairLaw g hg side).conditional | |
| (fun ij => halfWeightCode ij.1) k).mass) := | |
| entropy_sub_eq_conditional_sum _ _ _ _ _ | |
| (stageASplitLaw_ownWeight_marginal g hg side) | |
| (stageAPairLaw_leftWeight_marginal g hg side) | |
| theorem stageB_entropy_sub_pair_eq_conditional_sum | |
| (t : Shape) (ht : t ∈ positiveSecond) (side : Fin 3) : | |
| finiteEntropy (stageBSplitLaw t ht).mass - | |
| finiteEntropy (stageBPairLaw t ht side).mass = | |
| ∑ k, (stageBMarginalLaw t ht side).mass k * | |
| (finiteEntropy ((stageBSplitLaw t ht).conditional | |
| (stageBSplitWeight t ht side) k).mass - | |
| finiteEntropy ((stageBPairLaw t ht side).conditional | |
| (fun ij => statisticWeightCode ij.1) k).mass) := | |
| entropy_sub_eq_conditional_sum _ _ _ _ _ | |
| (stageBSplitLaw_ownWeight_marginal t ht side) | |
| (stageBPairLaw_leftWeight_marginal t ht side) | |
| end MatrixMultiplication.AllFieldPairConditionalEntropy | |
| end | |
| end OAI | |