openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /GroupedEntropyEmbedding.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.GroupedEntropyPartition | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointGroupedEntropyEmbedding | |
| open MatrixMultiplication.Foundation JointCompatibilityIncidence JointNativeGroupedEntropy | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {U V K L A : Type*} | |
| [Fintype U] [Fintype V] [Fintype K] [Fintype L] [Fintype A] | |
| omit [Fintype K] [Fintype L] in | |
| theorem groupMass_label_injective (j : K → L) (hj : Function.Injective j) | |
| (p : U → ℝ) (group : U → K) (k : K) : | |
| groupMass p (j ∘ group) (j k) = groupMass p group k := by | |
| simp only [groupMass, Function.comp_apply, hj.eq_iff] | |
| omit [Fintype K] [Fintype L] [Fintype A] in | |
| theorem groupLaw_label_injective (j : K → L) (hj : Function.Injective j) | |
| (p : U → ℝ) (group : U → K) (q : U → A → ℝ) (k : K) : | |
| groupLaw p (j ∘ group) q (j k) = groupLaw p group q k := by | |
| funext a | |
| simp only [groupLaw, groupMass_label_injective j hj, | |
| Function.comp_apply, hj.eq_iff] | |
| omit [Fintype K] [Fintype L] in | |
| theorem groupMass_label_not_mem_range (j : K → L) (p : U → ℝ) | |
| (group : U → K) (l : L) (hl : l ∉ Set.range j) : | |
| groupMass p (j ∘ group) l = 0 := by | |
| apply Finset.sum_eq_zero | |
| intro u _ | |
| exact ite_eq_right (fun equality => hl ⟨group u, equality⟩) | |
| theorem groupedEntropy_label_injective (j : K → L) (hj : Function.Injective j) | |
| (p : U → ℝ) (group : U → K) (q : U → A → ℝ) : | |
| groupedEntropy p (j ∘ group) q = groupedEntropy p group q := by | |
| unfold groupedEntropy | |
| symm | |
| apply Fintype.sum_of_injective j hj | |
| · intro l hl | |
| rw [groupMass_label_not_mem_range j p group l hl, zero_mul] | |
| · intro k | |
| rw [groupMass_label_injective j hj, groupLaw_label_injective j hj] | |
| def pushforwardWeight (p : U → ℝ) (f : U → V) (v : V) : ℝ := | |
| ∑ u, if f u = v then p u else 0 | |
| theorem sum_pushforwardWeight_mul (p : U → ℝ) (f : U → V) (test : V → ℝ) : | |
| (∑ v, pushforwardWeight p f v * test v) = ∑ u, p u * test (f u) := by | |
| simp only [pushforwardWeight, Finset.sum_mul] | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| simp only [ite_mul, zero_mul] | |
| simp | |
| omit [Fintype K] in | |
| theorem groupMass_pushforward (p : U → ℝ) (f : U → V) | |
| (group : V → K) (k : K) : | |
| groupMass (pushforwardWeight p f) group k = groupMass p (group ∘ f) k := by | |
| simpa only [groupMass, mul_ite, mul_one, mul_zero, Function.comp_apply] using | |
| (sum_pushforwardWeight_mul p f (fun v => if group v = k then 1 else 0)) | |
| omit [Fintype K] [Fintype A] in | |
| theorem groupMeasure_pushforward (p : U → ℝ) (f : U → V) | |
| (group : V → K) (q : V → A → ℝ) (k : K) : | |
| groupMeasure (pushforwardWeight p f) group q k = | |
| groupMeasure p (group ∘ f) (q ∘ f) k := by | |
| funext a | |
| simpa only [groupMeasure, mul_ite, mul_zero, Function.comp_apply] using | |
| (sum_pushforwardWeight_mul p f (fun v => if group v = k then q v a else 0)) | |
| omit [Fintype K] [Fintype A] in | |
| theorem groupLaw_pushforward (p : U → ℝ) (f : U → V) | |
| (group : V → K) (q : V → A → ℝ) (k : K) : | |
| groupLaw (pushforwardWeight p f) group q k = | |
| groupLaw p (group ∘ f) (q ∘ f) k := by | |
| funext a | |
| change groupMeasure (pushforwardWeight p f) group q k a / | |
| groupMass (pushforwardWeight p f) group k = | |
| groupMeasure p (group ∘ f) (q ∘ f) k a / groupMass p (group ∘ f) k | |
| rw [groupMeasure_pushforward, groupMass_pushforward] | |
| theorem groupedEntropy_pushforward (p : U → ℝ) (f : U → V) | |
| (group : V → K) (q : V → A → ℝ) : | |
| groupedEntropy (pushforwardWeight p f) group q = | |
| groupedEntropy p (group ∘ f) (q ∘ f) := by | |
| simp only [groupedEntropy, groupMass_pushforward, groupLaw_pushforward] | |
| theorem groupedEntropy_pushforward_of_agree (p : U → ℝ) (f : U → V) | |
| (group : V → K) (qU : U → A → ℝ) (qV : V → A → ℝ) | |
| (hq : ∀ u, qV (f u) = qU u) : | |
| groupedEntropy (pushforwardWeight p f) group qV = | |
| groupedEntropy p (group ∘ f) qU := by | |
| rw [groupedEntropy_pushforward] | |
| have he : qV ∘ f = qU := funext hq | |
| rw [he] | |
| theorem groupedEntropy_pushforward_embedding (p : U → ℝ) (f : U → V) | |
| (groupU : U → K) (groupV : V → L) (j : K → L) (hj : Function.Injective j) | |
| (hgroup : ∀ u, groupV (f u) = j (groupU u)) | |
| (qU : U → A → ℝ) (qV : V → A → ℝ) (hq : ∀ u, qV (f u) = qU u) : | |
| groupedEntropy (pushforwardWeight p f) groupV qV = groupedEntropy p groupU qU := by | |
| rw [groupedEntropy_pushforward_of_agree p f groupV qU qV hq] | |
| have hg : groupV ∘ f = j ∘ groupU := funext hgroup | |
| rw [hg, groupedEntropy_label_injective j hj] | |
| theorem sum_map_mass_mul (p : FiniteLaw U) (f : U → V) (test : V → ℝ) : | |
| (∑ v, (p.map f).mass v * test v) = ∑ u, p.mass u * test (f u) := by | |
| simpa only [FiniteLaw.map_mass, pushforwardWeight] using | |
| (sum_pushforwardWeight_mul p.mass f test) | |
| omit [Fintype K] in | |
| theorem groupMass_map (p : FiniteLaw U) (f : U → V) | |
| (group : V → K) (k : K) : | |
| groupMass (p.map f).mass group k = groupMass p.mass (group ∘ f) k := by | |
| simpa only [groupMass, mul_ite, mul_one, mul_zero, Function.comp_apply] using | |
| (sum_map_mass_mul p f (fun v => if group v = k then 1 else 0)) | |
| omit [Fintype K] [Fintype A] in | |
| theorem groupMeasure_map (p : FiniteLaw U) (f : U → V) | |
| (group : V → K) (q : V → A → ℝ) (k : K) : | |
| groupMeasure (p.map f).mass group q k = | |
| groupMeasure p.mass (group ∘ f) (q ∘ f) k := by | |
| funext a | |
| simpa only [groupMeasure, mul_ite, mul_zero, Function.comp_apply] using | |
| (sum_map_mass_mul p f (fun v => if group v = k then q v a else 0)) | |
| omit [Fintype K] [Fintype A] in | |
| theorem groupLaw_map (p : FiniteLaw U) (f : U → V) | |
| (group : V → K) (q : V → A → ℝ) (k : K) : | |
| groupLaw (p.map f).mass group q k = groupLaw p.mass (group ∘ f) (q ∘ f) k := by | |
| funext a | |
| change groupMeasure (p.map f).mass group q k a / groupMass (p.map f).mass group k = | |
| groupMeasure p.mass (group ∘ f) (q ∘ f) k a / groupMass p.mass (group ∘ f) k | |
| rw [groupMeasure_map, groupMass_map] | |
| theorem groupedEntropy_map (p : FiniteLaw U) (f : U → V) | |
| (group : V → K) (q : V → A → ℝ) : | |
| groupedEntropy (p.map f).mass group q = | |
| groupedEntropy p.mass (group ∘ f) (q ∘ f) := by | |
| simp only [groupedEntropy, groupMass_map, groupLaw_map] | |
| theorem groupedEntropy_map_of_agree (p : FiniteLaw U) (f : U → V) | |
| (group : V → K) (qU : U → A → ℝ) (qV : V → A → ℝ) | |
| (hq : ∀ u, qV (f u) = qU u) : | |
| groupedEntropy (p.map f).mass group qV = | |
| groupedEntropy p.mass (group ∘ f) qU := by | |
| rw [groupedEntropy_map] | |
| have he : qV ∘ f = qU := funext hq | |
| rw [he] | |
| theorem groupedEntropy_map_embedding (p : FiniteLaw U) (f : U → V) | |
| (groupU : U → K) (groupV : V → L) (j : K → L) (hj : Function.Injective j) | |
| (hgroup : ∀ u, groupV (f u) = j (groupU u)) | |
| (qU : U → A → ℝ) (qV : V → A → ℝ) (hq : ∀ u, qV (f u) = qU u) : | |
| groupedEntropy (p.map f).mass groupV qV = groupedEntropy p.mass groupU qU := by | |
| rw [groupedEntropy_map_of_agree p f groupV qU qV hq] | |
| have hg : groupV ∘ f = j ∘ groupU := funext hgroup | |
| rw [hg, groupedEntropy_label_injective j hj] | |
| end MatrixMultiplication.JointGroupedEntropyEmbedding | |
| end | |
| end OAI | |