openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /PairMixtureControls.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityControls | |
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityScaling | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointPairMixtureControls | |
| open JointCompatibilityControls JointCompatibilityScaling | |
| open scoped BigOperators | |
| attribute [local instance] Classical.propDecidable | |
| variable {C : Type*} [Fintype C] [DecidableEq C] | |
| {Shape A B : C → Type*} | |
| [∀ c, Fintype (Shape c)] [∀ c, DecidableEq (Shape c)] | |
| [∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)] | |
| [∀ c, Fintype (B c)] [∀ c, DecidableEq (B c)] | |
| def pairMixture (base : ∀ c, Shape c → ℕ) | |
| (left : ∀ c, Shape c → A c → ℝ) (right : ∀ c, Shape c → B c → ℝ) | |
| (c : C) (ab : A c × B c) : ℝ := | |
| (∑ u, (base c u : ℝ) * left c u ab.1 * right c u ab.2) / baseSize base c | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (B c)] in | |
| theorem sum_first_fiber (c : C) (q : A c × B c → ℝ) (a : A c) : | |
| (∑ ab : {ab : A c × B c // ab.1 = a}, q ab.val) = ∑ b, q (a, b) := by | |
| let e : B c ≃ {ab : A c × B c // ab.1 = a} := | |
| { toFun := fun b => ⟨(a, b), rfl⟩ | |
| invFun := fun ab => ab.val.2 | |
| left_inv := fun _ => rfl | |
| right_inv := fun ab => by apply Subtype.ext; exact Prod.ext ab.property.symm rfl } | |
| exact (e.sum_comp (fun ab => q ab.val)).symm | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (A c)] in | |
| theorem sum_second_fiber (c : C) (q : A c × B c → ℝ) (b : B c) : | |
| (∑ ab : {ab : A c × B c // ab.2 = b}, q ab.val) = ∑ a, q (a, b) := by | |
| let e : A c ≃ {ab : A c × B c // ab.2 = b} := | |
| { toFun := fun a => ⟨(a, b), rfl⟩ | |
| invFun := fun ab => ab.val.1 | |
| left_inv := fun _ => rfl | |
| right_inv := fun ab => by apply Subtype.ext; exact Prod.ext rfl ab.property.symm } | |
| exact (e.sum_comp (fun ab => q ab.val)).symm | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)] in | |
| theorem base_eq_zero_of_size_eq_zero (base : ∀ c, Shape c → ℕ) | |
| (c : C) (hz : baseSize base c = 0) (u : Shape c) : base c u = 0 := | |
| Nat.eq_zero_of_le_zero ((Finset.single_le_sum (fun _ _ => Nat.zero_le _) | |
| (Finset.mem_univ u)).trans_eq hz) | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)] | |
| [∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)] in | |
| theorem totalCenterCount_eq_zero_of_size_eq_zero (base : ∀ c, Shape c → ℕ) | |
| (law : ∀ c, Shape c → A c → ℝ) (c : C) (hz : baseSize base c = 0) (a : A c) : | |
| totalCenterCount base law c a = 0 := by | |
| simp only [totalCenterCount, base_eq_zero_of_size_eq_zero base c hz, | |
| Nat.cast_zero, zero_mul, Finset.sum_const_zero] | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)] | |
| [∀ c, DecidableEq (B c)] in | |
| theorem pairMixture_left_balance (base : ∀ c, Shape c → ℕ) | |
| (left : ∀ c, Shape c → A c → ℝ) (right : ∀ c, Shape c → B c → ℝ) | |
| (hnormalized : ∀ c u, 0 < base c u → ∑ b, right c u b = 1) | |
| (c : C) (a : A c) : | |
| (baseSize base c : ℝ) * | |
| (∑ ab : {ab : A c × B c // ab.1 = a}, pairMixture base left right c ab.val) = | |
| totalCenterCount base left c a := by | |
| rw [sum_first_fiber] | |
| simp only [pairMixture, ← Finset.sum_div] | |
| have hnum : (∑ b : B c, ∑ u : Shape c, | |
| (base c u : ℝ) * left c u a * right c u b) = totalCenterCount base left c a := by | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases hu : base c u = 0 | |
| · simp [hu] | |
| · rw [← Finset.mul_sum, hnormalized c u (Nat.pos_of_ne_zero hu), mul_one] | |
| rw [hnum] | |
| by_cases hz : baseSize base c = 0 | |
| · rw [totalCenterCount_eq_zero_of_size_eq_zero base left c hz a] | |
| simp | |
| · exact mul_div_cancel₀ _ (Nat.cast_ne_zero.mpr hz) | |
| omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (Shape c)] | |
| [∀ c, DecidableEq (A c)] in | |
| theorem pairMixture_right_balance (base : ∀ c, Shape c → ℕ) | |
| (left : ∀ c, Shape c → A c → ℝ) (right : ∀ c, Shape c → B c → ℝ) | |
| (hnormalized : ∀ c u, 0 < base c u → ∑ a, left c u a = 1) | |
| (c : C) (b : B c) : | |
| (baseSize base c : ℝ) * | |
| (∑ ab : {ab : A c × B c // ab.2 = b}, pairMixture base left right c ab.val) = | |
| totalCenterCount base right c b := by | |
| rw [sum_second_fiber] | |
| simp only [pairMixture, ← Finset.sum_div] | |
| have hnum : (∑ a : A c, ∑ u : Shape c, | |
| (base c u : ℝ) * left c u a * right c u b) = totalCenterCount base right c b := by | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases hu : base c u = 0 | |
| · simp [hu] | |
| · rw [← Finset.sum_mul, ← Finset.mul_sum, | |
| hnormalized c u (Nat.pos_of_ne_zero hu), mul_one] | |
| rw [hnum] | |
| by_cases hz : baseSize base c = 0 | |
| · rw [totalCenterCount_eq_zero_of_size_eq_zero base right c hz b] | |
| simp | |
| · exact mul_div_cancel₀ _ (Nat.cast_ne_zero.mpr hz) | |
| end MatrixMultiplication.JointPairMixtureControls | |
| end | |
| end OAI | |