openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /PairClassWindows.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.TypeCounts | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointPairClassWindows | |
| open MatrixMultiplication.Foundation | |
| attribute [local instance] Classical.propDecidable | |
| variable {I A C : Type*} [Fintype I] [Fintype A] [Fintype C] | |
| [DecidableEq I] [DecidableEq A] [DecidableEq C] | |
| abbrev Fiber (k : I → C) (c : C) := {i : I // k i = c} | |
| def fiberWord (z : I → A) (k : I → C) (c : C) : Fiber k c → A := | |
| fun i => z i.val | |
| omit [Fintype A] [Fintype C] [DecidableEq I] in | |
| theorem fiber_population (z : I → A) (k : I → C) (f : A → C) | |
| (hf : ∀ i, f (z i) = k i) (c : C) (a : A) : | |
| wordPopulation (fiberWord z k c) a = | |
| if f a = c then wordPopulation z a else 0 := by | |
| classical | |
| by_cases ha : f a = c | |
| · rw [ite_eq_left ha] | |
| unfold wordPopulation | |
| apply Fintype.card_congr | |
| exact | |
| { toFun := fun i => ⟨i.val.val, i.property⟩ | |
| invFun := fun i => ⟨⟨i.val, by rw [← hf i.val, i.property, ha]⟩, i.property⟩ | |
| left_inv := fun ⟨⟨i, hi⟩, hia⟩ => rfl | |
| right_inv := fun ⟨i, hia⟩ => rfl } | |
| · rw [ite_eq_right ha] | |
| let : IsEmpty {i : Fiber k c // fiberWord z k c i = a} := | |
| ⟨fun i => ha (calc | |
| f a = f (z i.val.val) := congrArg f i.property.symm | |
| _ = k i.val.val := hf _ | |
| _ = c := i.val.property)⟩ | |
| exact Fintype.card_eq_zero | |
| omit [Fintype C] [DecidableEq I] in | |
| theorem fiber_card (k : I → C) (c : C) : | |
| Fintype.card (Fiber k c) = wordPopulation k c := rfl | |
| def conditionalCenter (f : A → C) (c : C) (q : A → ℝ) (mass : ℝ) : A → ℝ := | |
| fun a => if f a = c then q a / mass else 0 | |
| omit [Fintype A] [Fintype C] [DecidableEq I] in | |
| theorem fiber_window (z : I → A) (k : I → C) (f : A → C) | |
| (hf : ∀ i, f (z i) = k i) (c : C) (q : A → ℝ) (mass η : ℝ) | |
| (hmass : 0 < mass) (hη : 0 ≤ η) | |
| (hclass : (Fintype.card (Fiber k c) : ℝ) = (Fintype.card I : ℝ) * mass) | |
| (hwindow : ∀ a, |(wordPopulation z a : ℝ) / Fintype.card I - q a| ≤ η) : | |
| ∀ a, |(wordPopulation (fiberWord z k c) a : ℝ) / Fintype.card (Fiber k c) - | |
| conditionalCenter f c q mass a| ≤ η / mass := by | |
| intro a | |
| rw [fiber_population z k f hf c a] | |
| by_cases ha : f a = c | |
| · simp only [ha, ite_true, conditionalCenter] | |
| rw [hclass] | |
| have hid : (wordPopulation z a : ℝ) / ((Fintype.card I : ℝ) * mass) - q a / mass = | |
| ((wordPopulation z a : ℝ) / Fintype.card I - q a) / mass := by | |
| rw [sub_div, div_div] | |
| rw [hid, abs_div, abs_of_pos hmass] | |
| exact div_le_div_of_nonneg_right (hwindow a) hmass.le | |
| · simp only [ha, ite_false, Nat.cast_zero, zero_div, conditionalCenter, sub_zero, abs_zero] | |
| exact div_nonneg hη hmass.le | |
| omit [Fintype A] [Fintype C] [DecidableEq I] in | |
| theorem fiber_count_window (z : I → A) (k : I → C) (f : A → C) | |
| (hf : ∀ i, f (z i) = k i) (c : C) (q : A → ℝ) (η : ℝ) (hη : 0 ≤ η) | |
| (hwindow : ∀ a, | |
| |(wordPopulation z a : ℝ) - (Fintype.card I : ℝ) * q a| ≤ (Fintype.card I : ℝ) * η) : | |
| ∀ a, |(wordPopulation (fiberWord z k c) a : ℝ) - | |
| (Fintype.card I : ℝ) * (if f a = c then q a else 0)| ≤ (Fintype.card I : ℝ) * η := by | |
| intro a | |
| rw [fiber_population z k f hf c a] | |
| by_cases ha : f a = c | |
| · simpa only [ha, ite_true] using hwindow a | |
| · simp only [ha, ite_false, Nat.cast_zero, mul_zero, sub_zero, abs_zero] | |
| exact mul_nonneg (Nat.cast_nonneg _) hη | |
| end MatrixMultiplication.JointPairClassWindows | |
| end | |
| end OAI | |