openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldParameters /LeafRates.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldParameters.Compatibility | |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexFiniteEntropy | |
| namespace OAI | |
| /-! Fixed rational distributions and their entropy and capacity formulas. -/ | |
| namespace MatrixMultiplication.AllFieldParameters | |
| open MatrixMultiplication.Foundation | |
| open scoped BigOperators | |
| noncomputable section | |
| theorem entropy_two_points {A : Type*} [Fintype A] [DecidableEq A] | |
| (a b : A) (hab : a ≠ b) (p q : ℝ) : | |
| finiteEntropy (fun i => if i = a then p else if i = b then q else 0) = | |
| entropyTerm p + entropyTerm q := by | |
| unfold finiteEntropy | |
| calc | |
| _ = ∑ i : A, ((if i = a then entropyTerm p else 0) + | |
| (if i = b then entropyTerm q else 0)) := by | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| by_cases hi : i = a | |
| · subst i | |
| simp [hab] | |
| · by_cases hj : i = b | |
| · subst i | |
| simp [hab.symm] | |
| · simp [hi, hj] | |
| _ = _ := by simp [Finset.sum_add_distrib] | |
| theorem weighted_two_points {A : Type*} [Fintype A] [DecidableEq A] | |
| (a b : A) (hab : a ≠ b) (p q : ℝ) (f : A → ℝ) : | |
| (∑ i : A, (if i = a then p else if i = b then q else 0) * f i) = | |
| p * f a + q * f b := by | |
| calc | |
| _ = ∑ i : A, ((if i = a then p * f a else 0) + | |
| (if i = b then q * f b else 0)) := by | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| by_cases hi : i = a | |
| · subst i | |
| simp [hab] | |
| · by_cases hj : i = b | |
| · subst i | |
| simp [hi] | |
| · simp [hi, hj] | |
| _ = _ := by simp [Finset.sum_add_distrib] | |
| def statisticRate (d : ℚ) (k : Fin 5) : ℝ := | |
| finiteEntropy (fun i : Fin 6 => (statisticLaw d k i : ℝ)) + | |
| ∑ i : Fin 6, (statisticLaw d k i : ℝ) * Real.log (multiplicity i : ℝ) | |
| def binaryEntropyRate (d : ℚ) : ℝ := | |
| entropyTerm (1 - (d : ℝ)) + entropyTerm (d : ℝ) + (d : ℝ) * Real.log 2 | |
| theorem statisticLaw_two_real (d : ℚ) (i : Fin 6) : | |
| (statisticLaw d 2 i : ℝ) = | |
| if i = 2 then 1 - (d : ℝ) else if i = 3 then (d : ℝ) else 0 := by | |
| by_cases hi : i = 2 | |
| · simp [statisticLaw, hi] | |
| · by_cases hj : i = 3 <;> simp [statisticLaw, hi, hj] | |
| theorem statisticRate_two (d : ℚ) : | |
| statisticRate d 2 = binaryEntropyRate d + 2 * (1 - (d : ℝ)) * Real.log 5 := by | |
| unfold statisticRate | |
| simp_rw [statisticLaw_two_real] | |
| rw [entropy_two_points (2 : Fin 6) 3 (by decide), | |
| weighted_two_points (2 : Fin 6) 3 (by decide)] | |
| have h25 : Real.log (25 : ℝ) = 2 * Real.log 5 := by | |
| have hp : (5 : ℝ) ^ 2 = 25 := by norm_num | |
| rw [← hp, Real.log_pow] | |
| norm_num | |
| change entropyTerm (1 - (d : ℝ)) + entropyTerm (d : ℝ) + | |
| ((1 - (d : ℝ)) * Real.log 25 + (d : ℝ) * Real.log 2) = _ | |
| rw [h25] | |
| unfold binaryEntropyRate | |
| ring | |
| theorem statisticRate_deterministic (d : ℚ) (k : Fin 5) (hk : k.val ≠ 2) : | |
| statisticRate d k = Real.log (multiplicity (singletonSlot k.val) : ℝ) := by | |
| have hmass (i : Fin 6) : (statisticLaw d k i : ℝ) = | |
| if i = singletonSlot k.val then 1 else 0 := by | |
| by_cases hi : i = singletonSlot k.val <;> simp [statisticLaw, hk, hi] | |
| unfold statisticRate finiteEntropy | |
| simp_rw [hmass] | |
| simp [entropyTerm, apply_ite] | |
| theorem statisticRate_zero (d : ℚ) : statisticRate d 0 = 0 := by | |
| rw [statisticRate_deterministic d 0 (by decide)] | |
| norm_num [singletonSlot, multiplicity] | |
| theorem statisticRate_one (d : ℚ) : statisticRate d 1 = Real.log 10 := by | |
| rw [statisticRate_deterministic d 1 (by decide)] | |
| norm_num [singletonSlot, multiplicity] | |
| theorem statisticRate_three (d : ℚ) : statisticRate d 3 = Real.log 10 := by | |
| rw [statisticRate_deterministic d 3 (by decide)] | |
| change Real.log (10 : ℝ) = Real.log 10 | |
| rfl | |
| theorem statisticRate_four (d : ℚ) : statisticRate d 4 = 0 := by | |
| rw [statisticRate_deterministic d 4 (by decide)] | |
| have hv : multiplicity (singletonSlot (4 : Fin 5).val) = 1 := by decide +kernel | |
| rw [hv] | |
| norm_num | |
| end | |
| end MatrixMultiplication.AllFieldParameters | |
| end OAI | |