openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /ProjectedGroupEntropy.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityRateLimit | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityControls | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.GroupedEntropyPartition | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointProjectedGroupEntropy | |
| open MatrixMultiplication.Foundation JointCompatibilityIncidence | |
| open JointCompatibilityScaling JointCompatibilityControls JointCompatibilityRateLimit | |
| open JointGroupedEntropyPartition | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| section FiniteGroups | |
| variable {U K A : Type*} [Fintype U] [Fintype K] [Fintype A] | |
| theorem groupMass_designated_some (p : U → ℝ) (d : U → Prop) (u : U) : | |
| groupMass p (designatedGroup d) (some u) = if d u then p u else 0 := by | |
| simp only [groupMass, designatedGroup_eq_some] | |
| by_cases hd : d u <;> simp [hd] | |
| theorem weighted_groupLaw_designated_some (p : U → ℝ) (d : U → Prop) | |
| (law : U → A → ℝ) (u : U) : | |
| groupMass p (designatedGroup d) (some u) * | |
| finiteEntropy (groupLaw p (designatedGroup d) law (some u)) = | |
| (if d u then p u else 0) * finiteEntropy (law u) := by | |
| by_cases hd : d u | |
| · rw [groupMass_designated_some, ite_eq_left hd] | |
| by_cases hp : p u = 0 | |
| · simp [hp] | |
| · have hl : groupLaw p (designatedGroup d) law (some u) = law u := by | |
| funext a | |
| simp [groupLaw, groupMass_designated_some, designatedGroup_eq_some, hd, hp] | |
| rw [hl] | |
| · rw [groupMass_designated_some, ite_eq_right hd] | |
| simp | |
| theorem groupedEntropy_counts_eq_total_mul (n : U → ℕ) (group : U → K) | |
| (law : U → A → ℝ) : | |
| groupedEntropy (fun u => (n u : ℝ)) group law = | |
| (∑ u, n u : ℕ) * | |
| groupedEntropy (fun u => (n u : ℝ) / (∑ v, n v : ℕ)) group law := by | |
| by_cases hz : (∑ u, n u) = 0 | |
| · have hn (u : U) : n u = 0 := by | |
| have hle : n u ≤ ∑ v, n v := | |
| Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ u) | |
| rw [hz] at hle | |
| exact Nat.eq_zero_of_le_zero hle | |
| simp [hn, groupedEntropy, groupMass] | |
| · have hN : ((∑ u, n u : ℕ) : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hz | |
| have hf : (fun u => (n u : ℝ)) = | |
| fun u => ((∑ v, n v : ℕ) : ℝ) * ((n u : ℝ) / (∑ v, n v : ℕ)) := by | |
| funext u | |
| exact (mul_div_cancel₀ _ hN).symm | |
| calc | |
| _ = groupedEntropy | |
| (fun u => ((∑ v, n v : ℕ) : ℝ) * ((n u : ℝ) / (∑ v, n v : ℕ))) | |
| group law := congrArg (fun p => groupedEntropy p group law) hf | |
| _ = _ := groupedEntropy_mul _ _ _ _ | |
| end FiniteGroups | |
| variable {C : Type*} [Fintype C] [DecidableEq C] | |
| {Shape A : C → Type*} | |
| [∀ c, Fintype (Shape c)] [∀ c, DecidableEq (Shape c)] | |
| [∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)] | |
| omit [Fintype C] [DecidableEq C] in | |
| theorem baseGroupSize_eq_groupMass (base : ∀ c, Shape c → ℕ) | |
| (d : ∀ c, Shape c → Prop) (g : Σ c, Option (Shape c)) : | |
| (baseGroupSize base d g : ℝ) = | |
| groupMass (fun u => (base g.1 u : ℝ)) (designatedGroup (d g.1)) g.2 := by | |
| calc | |
| (baseGroupSize base d g : ℝ) = | |
| ∑ u ∈ Finset.univ.filter (fun u => designatedGroup (d g.1) u = g.2), | |
| (base g.1 u : ℝ) := by | |
| rw [baseGroupSize, Nat.cast_sum] | |
| exact (Finset.sum_subtype | |
| (Finset.univ.filter (fun u => designatedGroup (d g.1) u = g.2)) | |
| (by simp) (fun u => (base g.1 u : ℝ))).symm | |
| _ = ∑ u, if designatedGroup (d g.1) u = g.2 then (base g.1 u : ℝ) else 0 := by | |
| rw [Finset.sum_filter] | |
| _ = _ := by | |
| unfold groupMass | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases hu : designatedGroup (d g.1) u = g.2 <;> simp only [hu, ite_true, ite_false] | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)] | |
| [∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)] in | |
| theorem residualCenter_eq_groupLaw (base : ∀ c, Shape c → ℕ) | |
| (d : ∀ c, Shape c → Prop) (law : ∀ c, Shape c → A c → ℝ) (c : C) : | |
| residualCenter base d law c = | |
| groupLaw (fun u => (base c u : ℝ)) (designatedGroup (d c)) (law c) none := by | |
| funext a | |
| simp only [residualCenter, residualBase, Nat.cast_sum, groupLaw, groupMass, | |
| designatedGroup_eq_none] | |
| congr 1 | |
| · rw [← Finset.sum_filter] | |
| exact (Finset.sum_subtype (Finset.univ.filter (fun u => ¬ d c u)) | |
| (by simp) (fun u => (base c u : ℝ) * law c u a)).symm | |
| · rw [← Finset.sum_filter] | |
| exact (Finset.sum_subtype (Finset.univ.filter (fun u => ¬ d c u)) | |
| (by simp) (fun u => (base c u : ℝ))).symm | |
| omit [DecidableEq C] [∀ c, DecidableEq (A c)] in | |
| theorem projectedReferenceEntropy_eq_sum_counts_groupedEntropy | |
| (base : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop) | |
| (law : ∀ c, Shape c → A c → ℝ) : | |
| projectedReferenceEntropy base d (groupCap base d law 0) = | |
| ∑ c, groupedEntropy (fun u => (base c u : ℝ)) | |
| (designatedGroup (d c)) (law c) := by | |
| simp only [projectedReferenceEntropy, Fintype.sum_sigma, groupCap, add_zero, | |
| groupedEntropy] | |
| apply Finset.sum_congr rfl | |
| intro c _ | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| rw [baseGroupSize_eq_groupMass] | |
| cases k with | |
| | none => | |
| change _ * finiteEntropy (residualCenter base d law c) = _ | |
| rw [residualCenter_eq_groupLaw] | |
| | some u => | |
| change _ * finiteEntropy (law c u) = _ | |
| rw [weighted_groupLaw_designated_some, groupMass_designated_some] | |
| omit [DecidableEq C] [∀ c, DecidableEq (A c)] in | |
| theorem projectedReferenceEntropy_eq_sum_weighted_groupedEntropy | |
| (base : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop) | |
| (law : ∀ c, Shape c → A c → ℝ) : | |
| projectedReferenceEntropy base d (groupCap base d law 0) = | |
| ∑ c, (baseSize base c : ℝ) * | |
| groupedEntropy (fun u => (base c u : ℝ) / (baseSize base c : ℝ)) | |
| (designatedGroup (d c)) (law c) := by | |
| rw [projectedReferenceEntropy_eq_sum_counts_groupedEntropy] | |
| apply Finset.sum_congr rfl | |
| intro c _ | |
| exact groupedEntropy_counts_eq_total_mul (base c) _ _ | |
| end MatrixMultiplication.JointProjectedGroupEntropy | |
| end | |
| end OAI | |