openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Profiles /ProfileMarginalContinuity.lean
| import Mathlib.MeasureTheory.Group.Arithmetic | |
| import OAI.Combinatorics.BalancedRyser.Profiles.Profile | |
| import OAI.Combinatorics.BalancedRyser.Profiles.ProfileContinuity | |
| namespace OAI | |
| open MeasureTheory | |
| open scoped Topology | |
| namespace Balanced.ProfileContinuity | |
| theorem row_sum_eq_selected_sum (p : Profile) (hp : Summable p) (i : ℕ) : | |
| (∑' j, p (i, j)) = ∑' ij, p ij * (if ij.1 = i then 1 else 0) := by | |
| classical | |
| have hs := summable_mul_of_abs_le_one hp | |
| (w := fun ij => if ij.1 = i then 1 else 0) (by intro ij; split_ifs <;> norm_num) | |
| rw [hs.tsum_prod] | |
| simp only [mul_ite, mul_one, mul_zero] | |
| conv_rhs => | |
| arg 1 | |
| ext k | |
| rw [show (∑' j, if k = i then p (k, j) else 0) = | |
| if k = i then ∑' j, p (k, j) else 0 by | |
| by_cases hk : k = i <;> simp [hk]] | |
| simp | |
| theorem continuous_profile_row_sum {S : Set Profile} | |
| (hnonneg : ∀ p ∈ S, ∀ ij, 0 ≤ p ij) | |
| (hsum : ∀ p ∈ S, HasSum p 1) (i : ℕ) : | |
| Continuous (fun p : S => ∑' j, p.val (i, j)) := by | |
| classical | |
| have hc : Continuous (fun p : S => | |
| ∑' ij, p.val ij * (if ij.1 = i then 1 else 0)) := by | |
| apply continuous_tsum_mul_of_probability_mass | |
| · exact fun p => hnonneg p.val p.property | |
| · exact fun p => hsum p.val p.property | |
| · intro ij | |
| exact (continuous_apply ij).comp continuous_subtype_val | |
| · intro ij | |
| exact continuous_const | |
| · intro p ij | |
| split_ifs <;> norm_num | |
| convert hc using 1 | |
| funext p | |
| exact row_sum_eq_selected_sum p.val (hsum p.val p.property).summable i | |
| theorem continuous_boundedProfile_row_sum (M : ℝ) (i : ℕ) : | |
| Continuous (fun p : {p : Profile | IsBoundedProfile M p} => | |
| ∑' j : ℕ, p.val (i, j)) := | |
| continuous_profile_row_sum (fun _ hp => hp.1) (fun _ hp => hp.2.1) i | |
| theorem continuous_profile_column_sum {S : Set Profile} | |
| (hnonneg : ∀ p ∈ S, ∀ ij, 0 ≤ p ij) | |
| (hsum : ∀ p ∈ S, HasSum p 1) (j : ℕ) : | |
| Continuous (fun p : S => ∑' i, p.val (i, j)) := by | |
| classical | |
| have hc : Continuous (fun p : S => | |
| ∑' ij, p.val ij * (if ij.2 = j then 1 else 0)) := by | |
| apply continuous_tsum_mul_of_probability_mass | |
| · exact fun p => hnonneg p.val p.property | |
| · exact fun p => hsum p.val p.property | |
| · intro ij | |
| exact (continuous_apply ij).comp continuous_subtype_val | |
| · intro ij | |
| exact continuous_const | |
| · intro p ij | |
| split_ifs <;> norm_num | |
| convert hc using 1 | |
| funext p | |
| have hs := summable_mul_of_abs_le_one (hsum p.val p.property).summable | |
| (w := fun ij => if ij.2 = j then 1 else 0) (by intro ij; split_ifs <;> norm_num) | |
| rw [hs.tsum_prod] | |
| simp | |
| theorem continuous_boundedProfile_column_sum (M : ℝ) (j : ℕ) : | |
| Continuous (fun p : {p : Profile | IsBoundedProfile M p} => | |
| ∑' i : ℕ, p.val (i, j)) := | |
| continuous_profile_column_sum (fun _ hp => hp.1) (fun _ hp => hp.2.1) j | |
| theorem measurable_profile_row_sum (i : ℕ) : | |
| Measurable (fun p : Profile => ∑' j : ℕ, p (i, j)) := | |
| Measurable.tsum (fun j => measurable_pi_apply (i, j)) | |
| theorem measurable_profile_column_sum (j : ℕ) : | |
| Measurable (fun p : Profile => ∑' i : ℕ, p (i, j)) := | |
| Measurable.tsum (fun i => measurable_pi_apply (i, j)) | |
| end Balanced.ProfileContinuity | |
| end OAI | |