openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /GroupedEntropyPartition.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.NativeGroupedEntropy | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointGroupedEntropyPartition | |
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory | |
| open JointNativeGroupedEntropy | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| section FiniteGroups | |
| variable {U K A : Type*} [Fintype U] [Fintype K] [Fintype A] | |
| omit [Fintype K] in | |
| theorem groupMass_mul (s : ℝ) (p : U → ℝ) (group : U → K) (k : K) : | |
| JointCompatibilityIncidence.groupMass (fun u => s * p u) group k = | |
| s * JointCompatibilityIncidence.groupMass p group k := by | |
| unfold JointCompatibilityIncidence.groupMass | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases hu : group u = k <;> simp [hu] | |
| omit [Fintype K] [Fintype A] in | |
| theorem groupMeasure_mul (s : ℝ) (p : U → ℝ) (group : U → K) | |
| (q : U → A → ℝ) (k : K) : | |
| groupMeasure (fun u => s * p u) group q k = | |
| fun a => s * groupMeasure p group q k a := by | |
| funext a | |
| unfold groupMeasure | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases hu : group u = k <;> simp [hu, mul_assoc] | |
| omit [Fintype K] [Fintype A] in | |
| theorem groupLaw_mul_of_ne_zero (s : ℝ) (hs : s ≠ 0) (p : U → ℝ) | |
| (group : U → K) (q : U → A → ℝ) (k : K) : | |
| JointCompatibilityIncidence.groupLaw (fun u => s * p u) group q k = | |
| JointCompatibilityIncidence.groupLaw p group q k := by | |
| funext a | |
| change groupMeasure (fun u => s * p u) group q k a / | |
| JointCompatibilityIncidence.groupMass (fun u => s * p u) group k = | |
| groupMeasure p group q k a / JointCompatibilityIncidence.groupMass p group k | |
| rw [groupMeasure_mul, groupMass_mul] | |
| exact mul_div_mul_left _ _ hs | |
| theorem groupedEntropy_mul (s : ℝ) (p : U → ℝ) (group : U → K) | |
| (q : U → A → ℝ) : | |
| JointCompatibilityIncidence.groupedEntropy (fun u => s * p u) group q = | |
| s * JointCompatibilityIncidence.groupedEntropy p group q := by | |
| by_cases hs : s = 0 | |
| · simp [hs, JointCompatibilityIncidence.groupedEntropy, | |
| JointCompatibilityIncidence.groupMass] | |
| · unfold JointCompatibilityIncidence.groupedEntropy | |
| simp only [groupMass_mul, groupLaw_mul_of_ne_zero s hs, mul_assoc, Finset.mul_sum] | |
| omit [Fintype A] in | |
| theorem groupMeasure_designated_some (p : U → ℝ) (d : U → Prop) | |
| (q : U → A → ℝ) (u : U) : | |
| groupMeasure p (JointCompatibilityIncidence.designatedGroup d) q (some u) = | |
| fun a => if d u then p u * q u a else 0 := by | |
| funext a | |
| simp only [groupMeasure, JointCompatibilityIncidence.designatedGroup_eq_some] | |
| by_cases hu : d u <;> simp [hu] | |
| omit [Fintype A] in | |
| theorem groupMeasure_designated_none (p : U → ℝ) (d : U → Prop) | |
| (q : U → A → ℝ) : | |
| groupMeasure p (JointCompatibilityIncidence.designatedGroup d) q none = | |
| fun a => ∑ u, if ¬ d u then p u * q u a else 0 := by | |
| funext a | |
| simp only [groupMeasure, JointCompatibilityIncidence.designatedGroup_eq_none] | |
| theorem groupedEntropy_designated (p : U → ℝ) (d : U → Prop) | |
| (q : U → A → ℝ) (hp : ∀ u, 0 ≤ p u) (hq : ∀ u, ∑ a, q u a = 1) : | |
| JointCompatibilityIncidence.groupedEntropy p | |
| (JointCompatibilityIncidence.designatedGroup d) q = | |
| (∑ u, if d u then p u * homogeneousEntropy (q u) else 0) + | |
| homogeneousEntropy (fun a => ∑ u, if ¬ d u then p u * q u a else 0) := by | |
| rw [groupedEntropy_eq_sum_homogeneous p _ q hp hq, Fintype.sum_option, | |
| groupMeasure_designated_none, add_comm] | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| rw [groupMeasure_designated_some] | |
| by_cases hu : d u | |
| · simp only [hu, ite_true] | |
| exact homogeneousEntropy_mul_normalized (p u) (q u) (hq u) | |
| · simp only [hu, ite_false, homogeneousEntropy_zero] | |
| theorem groupedEntropy_partition (p : U → ℝ) (d : U → Prop) (weight : U → K) | |
| (q : U → A → ℝ) (hp : ∀ u, 0 ≤ p u) (hq : ∀ u, ∑ a, q u a = 1) : | |
| JointCompatibilityIncidence.groupedEntropy p (singletonResidualGroup d weight) q = | |
| ∑ k, JointCompatibilityIncidence.groupedEntropy | |
| (fun u => if weight u = k then p u else 0) | |
| (JointCompatibilityIncidence.designatedGroup d) q := by | |
| have hclass (k : K) := groupedEntropy_designated | |
| (fun u => if weight u = k then p u else 0) d q | |
| (fun u => by split_ifs; exact hp u; exact le_rfl) hq | |
| rw [groupedEntropy_singletons_residuals p d weight q hp hq] | |
| simp_rw [hclass] | |
| rw [Finset.sum_add_distrib, Finset.sum_comm] | |
| congr 1 | |
| · apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases hu : d u <;> simp [hu, ite_mul] | |
| · apply Finset.sum_congr rfl | |
| intro k _ | |
| congr 1 | |
| funext a | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases hd : d u <;> by_cases hw : weight u = k <;> simp [hd, hw] | |
| theorem groupedEntropy_conditional_partition (p : FiniteLaw U) (d : U → Prop) | |
| (weight : U → K) (q : U → A → ℝ) (hq : ∀ u, ∑ a, q u a = 1) : | |
| JointCompatibilityIncidence.groupedEntropy p.mass (singletonResidualGroup d weight) q = | |
| ∑ k, (p.map weight).mass k * | |
| JointCompatibilityIncidence.groupedEntropy (p.conditional weight k).mass | |
| (JointCompatibilityIncidence.designatedGroup d) q := by | |
| rw [groupedEntropy_partition p.mass d weight q p.nonneg hq] | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| have hm : (fun u => if weight u = k then p.mass u else 0) = | |
| fun u => (p.map weight).mass k * (p.conditional weight k).mass u := by | |
| funext u | |
| exact (p.map_mass_mul_conditional weight k u).symm | |
| rw [hm, groupedEntropy_mul] | |
| end FiniteGroups | |
| section NativeSupport | |
| variable {A : Type*} [Fintype A] | |
| theorem native_groupedEntropy_eq_sum_masked (support : List Shape) (p : Shape → ℝ) | |
| (priority : Placement) (position : Fin 3) (q : Shape → A → ℝ) | |
| (hbound : ∀ i : SupportIndex support, support[i.val] (priority position) < 17) | |
| (hp : ∀ i : SupportIndex support, 0 ≤ p support[i.val]) | |
| (hq : ∀ i : SupportIndex support, ∑ a, q support[i.val] a = 1) : | |
| AllFieldNativeCapacity.groupedEntropy support p priority position q = | |
| ∑ k : Fin 17, JointCompatibilityIncidence.groupedEntropy | |
| (fun i : SupportIndex support => | |
| if ownWeight support priority position hbound i = k then p support[i.val] else 0) | |
| (JointCompatibilityIncidence.designatedGroup | |
| (fun i => AllFieldNativeCapacity.designated priority position support[i.val])) | |
| (fun i => q support[i.val]) := by | |
| have hpart := groupedEntropy_partition (fun i : SupportIndex support => p support[i.val]) | |
| (fun i => AllFieldNativeCapacity.designated priority position support[i.val]) | |
| (ownWeight support priority position hbound) (fun i => q support[i.val]) hp hq | |
| refine (groupedEntropy_eq_native support p priority position q hbound hp hq).symm.trans | |
| (hpart.trans ?_) | |
| apply Finset.sum_congr rfl | |
| intro k _ | |
| apply congrArg (fun weights : SupportIndex support → ℝ => | |
| JointCompatibilityIncidence.groupedEntropy weights | |
| (JointCompatibilityIncidence.designatedGroup | |
| (fun i => AllFieldNativeCapacity.designated priority position support[i.val])) | |
| (fun i => q support[i.val])) | |
| funext i | |
| by_cases hi : ownWeight support priority position hbound i = k <;> simp only [hi, ite_true, ite_false] | |
| theorem native_groupedEntropy_eq_sum_conditional (support : List Shape) (p : Shape → ℝ) | |
| (priority : Placement) (position : Fin 3) (q : Shape → A → ℝ) | |
| (hbound : ∀ i : SupportIndex support, support[i.val] (priority position) < 17) | |
| (law : FiniteLaw (SupportIndex support)) | |
| (hlaw : ∀ i : SupportIndex support, law.mass i = p support[i.val]) | |
| (hq : ∀ i : SupportIndex support, ∑ a, q support[i.val] a = 1) : | |
| AllFieldNativeCapacity.groupedEntropy support p priority position q = | |
| ∑ k : Fin 17, (law.map (ownWeight support priority position hbound)).mass k * | |
| JointCompatibilityIncidence.groupedEntropy | |
| (law.conditional (ownWeight support priority position hbound) k).mass | |
| (JointCompatibilityIncidence.designatedGroup | |
| (fun i => AllFieldNativeCapacity.designated priority position support[i.val])) | |
| (fun i => q support[i.val]) := by | |
| have hp (i : SupportIndex support) : 0 ≤ p support[i.val] := by | |
| rw [← hlaw i] | |
| exact law.nonneg i | |
| have hm : law.mass = fun i : SupportIndex support => p support[i.val] := funext hlaw | |
| have hn : JointCompatibilityIncidence.groupedEntropy law.mass | |
| (nativeGroup support priority position hbound) (fun i => q support[i.val]) = | |
| AllFieldNativeCapacity.groupedEntropy support p priority position q := by | |
| rw [hm] | |
| exact groupedEntropy_eq_native support p priority position q hbound hp hq | |
| exact hn.symm.trans (groupedEntropy_conditional_partition law _ _ _ hq) | |
| end NativeSupport | |
| end MatrixMultiplication.JointGroupedEntropyPartition | |
| end | |
| end OAI | |