openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Recovery /PermutationIndicatorMoments.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.FiniteAverage | |
| import Mathlib.GroupTheory.Perm.Finite | |
| import Mathlib.Algebra.Group.Units.Equiv | |
| import Mathlib.Data.Fintype.EquivFin | |
| import Mathlib.Data.Fintype.Perm | |
| namespace OAI | |
| /-! Finite orbit symmetries, masks and exact recovery operations. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.PermutationMatching | |
| open scoped BigOperators | |
| variable {P : Type*} [DecidableEq P] | |
| def membership (B : Finset P) (p : Equiv.Perm P) (i : P) : ℝ := | |
| if p i ∈ B then 1 else 0 | |
| @[simp] theorem membership_sq (B : Finset P) (p : Equiv.Perm P) (i : P) : | |
| membership B p i * membership B p i = membership B p i := by | |
| simp only [membership] | |
| split_ifs <;> norm_num | |
| theorem membership_nonneg (B : Finset P) (p : Equiv.Perm P) (i : P) : | |
| 0 ≤ membership B p i := by | |
| simp only [membership] | |
| split_ifs <;> norm_num | |
| theorem membership_le_one (B : Finset P) (p : Equiv.Perm P) (i : P) : | |
| membership B p i ≤ 1 := by | |
| simp only [membership] | |
| split_ifs <;> norm_num | |
| variable [Fintype P] | |
| theorem sum_membership (B : Finset P) (p : Equiv.Perm P) : | |
| (∑ i : P, membership B p i) = (B.card : ℝ) := by | |
| unfold membership | |
| rw [Equiv.sum_comp p (fun i => if i ∈ B then (1 : ℝ) else 0)] | |
| simp | |
| theorem sum_perm_right (f : Equiv.Perm P → ℝ) (q : Equiv.Perm P) : | |
| (∑ p : Equiv.Perm P, f (p * q)) = ∑ p : Equiv.Perm P, f p := by | |
| exact Equiv.sum_comp (Equiv.mulRight q) f | |
| theorem sum_membership_position_eq (B : Finset P) (i j : P) : | |
| (∑ p : Equiv.Perm P, membership B p i) = | |
| ∑ p : Equiv.Perm P, membership B p j := by | |
| calc | |
| _ = ∑ p : Equiv.Perm P, membership B (p * Equiv.swap i j) j := by | |
| simp [membership, Equiv.Perm.mul_apply] | |
| _ = _ := sum_perm_right (fun p => membership B p j) _ | |
| theorem card_mul_sum_membership (B : Finset P) (i : P) : | |
| (Fintype.card P : ℝ) * (∑ p : Equiv.Perm P, membership B p i) = | |
| (Fintype.card (Equiv.Perm P) : ℝ) * B.card := by | |
| calc | |
| _ = ∑ j : P, ∑ p : Equiv.Perm P, membership B p i := by simp | |
| _ = ∑ j : P, ∑ p : Equiv.Perm P, membership B p j := by | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| exact sum_membership_position_eq B i j | |
| _ = ∑ p : Equiv.Perm P, ∑ j : P, membership B p j := Finset.sum_comm | |
| _ = _ := by simp [sum_membership] | |
| theorem average_membership [Nonempty P] (B : Finset P) (i : P) : | |
| average (fun p : Equiv.Perm P => membership B p i) = | |
| (B.card : ℝ) / Fintype.card P := by | |
| have hn : (Fintype.card P : ℝ) ≠ 0 := by | |
| exact_mod_cast Fintype.card_ne_zero | |
| have hp : (Fintype.card (Equiv.Perm P) : ℝ) ≠ 0 := by | |
| exact_mod_cast Fintype.card_ne_zero | |
| unfold average | |
| apply (div_eq_div_iff hp hn).2 | |
| simpa only [mul_comm] using card_mul_sum_membership B i | |
| theorem sum_membership_mul_position_eq (B : Finset P) {i j k : P} | |
| (hij : i ≠ j) (hik : i ≠ k) : | |
| (∑ p : Equiv.Perm P, membership B p i * membership B p j) = | |
| ∑ p : Equiv.Perm P, membership B p i * membership B p k := by | |
| calc | |
| _ = ∑ p : Equiv.Perm P, | |
| membership B (p * Equiv.swap j k) i * | |
| membership B (p * Equiv.swap j k) k := by | |
| simp [membership, Equiv.Perm.mul_apply, | |
| Equiv.swap_apply_of_ne_of_ne hij hik] | |
| _ = _ := sum_perm_right (fun p => membership B p i * membership B p k) _ | |
| theorem card_sub_one_mul_sum_membership_mul (B : Finset P) {i j : P} | |
| (hij : i ≠ j) : | |
| ((Fintype.card P : ℝ) - 1) * | |
| (∑ p : Equiv.Perm P, membership B p i * membership B p j) = | |
| ((B.card : ℝ) - 1) * (∑ p : Equiv.Perm P, membership B p i) := by | |
| have hcard : (((Finset.univ : Finset P).erase i).card : ℝ) = | |
| (Fintype.card P : ℝ) - 1 := by | |
| have h := congrArg (fun n : ℕ => (n : ℝ)) | |
| (Finset.card_erase_add_one (Finset.mem_univ i)) | |
| simp only [Nat.cast_add, Nat.cast_one, Finset.card_univ] at h | |
| linarith | |
| have herase : | |
| (∑ k ∈ (Finset.univ : Finset P).erase i, | |
| ∑ p : Equiv.Perm P, membership B p i * membership B p k) = | |
| ((Fintype.card P : ℝ) - 1) * | |
| (∑ p : Equiv.Perm P, membership B p i * membership B p j) := by | |
| calc | |
| _ = ∑ k ∈ (Finset.univ : Finset P).erase i, | |
| ∑ p : Equiv.Perm P, membership B p i * membership B p j := by | |
| apply Finset.sum_congr rfl | |
| intro k hk | |
| exact sum_membership_mul_position_eq B (Finset.mem_erase.mp hk).1.symm hij | |
| _ = _ := by simp only [Finset.sum_const, nsmul_eq_mul, hcard] | |
| have htotal : | |
| (∑ k : P, ∑ p : Equiv.Perm P, membership B p i * membership B p k) = | |
| (B.card : ℝ) * (∑ p : Equiv.Perm P, membership B p i) := by | |
| rw [Finset.sum_comm] | |
| simp_rw [← Finset.mul_sum, sum_membership] | |
| rw [← Finset.sum_mul, mul_comm] | |
| have hsplit := Finset.sum_erase_add (Finset.univ : Finset P) | |
| (fun k => ∑ p : Equiv.Perm P, membership B p i * membership B p k) | |
| (Finset.mem_univ i) | |
| rw [herase, htotal] at hsplit | |
| simp only [membership_sq] at hsplit | |
| nlinarith | |
| theorem average_membership_mul_of_ne (B : Finset P) {i j : P} (hij : i ≠ j) : | |
| average (fun p : Equiv.Perm P => membership B p i * membership B p j) = | |
| (B.card : ℝ) * ((B.card : ℝ) - 1) / | |
| ((Fintype.card P : ℝ) * ((Fintype.card P : ℝ) - 1)) := by | |
| have hnlarge : 1 < Fintype.card P := Fintype.one_lt_card_iff.mpr ⟨i, j, hij⟩ | |
| have hnlarge' : (1 : ℝ) < Fintype.card P := by exact_mod_cast hnlarge | |
| have hn : (Fintype.card P : ℝ) ≠ 0 := by linarith | |
| have hn1 : (Fintype.card P : ℝ) - 1 ≠ 0 := by linarith | |
| have hp : (Fintype.card (Equiv.Perm P) : ℝ) ≠ 0 := by | |
| exact_mod_cast Fintype.card_ne_zero | |
| unfold average | |
| apply (div_eq_div_iff hp (mul_ne_zero hn hn1)).2 | |
| calc | |
| _ = (Fintype.card P : ℝ) * (((Fintype.card P : ℝ) - 1) * | |
| (∑ p : Equiv.Perm P, membership B p i * membership B p j)) := by ring | |
| _ = (Fintype.card P : ℝ) * (((B.card : ℝ) - 1) * | |
| (∑ p : Equiv.Perm P, membership B p i)) := by | |
| rw [card_sub_one_mul_sum_membership_mul B hij] | |
| _ = ((B.card : ℝ) - 1) * ((Fintype.card P : ℝ) * | |
| (∑ p : Equiv.Perm P, membership B p i)) := by ring | |
| _ = ((B.card : ℝ) - 1) * ((Fintype.card (Equiv.Perm P) : ℝ) * B.card) := by | |
| rw [card_mul_sum_membership B i] | |
| _ = _ := by ring | |
| end MatrixMultiplication.PermutationMatching | |
| end | |
| end OAI | |