| import Mathlib.Algebra.BigOperators.Group.Finset.Sigma |
| import Mathlib.Algebra.BigOperators.Ring.Finset |
| import Mathlib.Basic.Real.Basic |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| /-! Symmetric double and quadruple sums over unordered pairs. -/ |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
| namespace Paper170 |
|
|
| def upperPairSum {α : Type*} [Fintype α] [LinearOrder α] (f : α → α → ℝ) : ℝ := |
| ∑ i, ∑ p, if i < p then f i p else 0 |
|
|
| theorem upperPairSum_mul {α : Type*} [Fintype α] [LinearOrder α] |
| (r : ℝ) (f : α → α → ℝ) : |
| upperPairSum (fun i p => r * f i p) = r * upperPairSum f := by |
| simp only [upperPairSum, Finset.mul_sum, mul_ite, mul_zero] |
|
|
| theorem sum_symmetric_pair {α : Type*} [Fintype α] [LinearOrder α] |
| (f : α → α → ℝ) (hsym : ∀ i p, f p i = f i p) (hdiag : ∀ i, f i i = 0) : |
| (∑ i, ∑ p, f i p) = 2 * upperPairSum f := by |
| have hpoint (i p : α) : f i p = |
| (if i < p then f i p else 0) + (if p < i then f i p else 0) := by |
| rcases lt_trichotomy i p with h | rfl | h |
| · simp [h, not_lt_of_ge h.le] |
| · simp [hdiag] |
| · simp [h, not_lt_of_ge h.le] |
| have hsum : (∑ i, ∑ p, f i p) = |
| upperPairSum f + ∑ i, ∑ p, if p < i then f i p else 0 := by |
| calc |
| _ = ∑ i, ∑ p, ((if i < p then f i p else 0) + |
| (if p < i then f i p else 0)) := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro p _ |
| exact hpoint i p |
| _ = _ := by simp only [upperPairSum, Finset.sum_add_distrib] |
| rw [hsum, Finset.sum_comm] |
| have he : (∑ p, ∑ i, if p < i then f i p else 0) = upperPairSum f := by |
| unfold upperPairSum |
| apply Finset.sum_congr rfl |
| intro p _ |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [hsym] |
| rw [he] |
| ring |
|
|
| theorem sum_symmetric_quad {α : Type*} [Fintype α] [LinearOrder α] |
| (f : α → α → α → α → ℝ) |
| (hfirst : ∀ i p k q, f p i k q = f i p k q) |
| (hsecond : ∀ i p k q, f i p q k = f i p k q) |
| (hdiag₁ : ∀ i k q, f i i k q = 0) (hdiag₂ : ∀ i p k, f i p k k = 0) : |
| (∑ i, ∑ p, ∑ k, ∑ q, f i p k q) = |
| 4 * upperPairSum (fun i p => upperPairSum (f i p)) := by |
| have hs (i p : α) : (∑ k, ∑ q, f p i k q) = ∑ k, ∑ q, f i p k q := by |
| simp only [hfirst] |
| have hd (i : α) : (∑ k, ∑ q, f i i k q) = 0 := by simp [hdiag₁] |
| rw [sum_symmetric_pair _ hs hd] |
| have he : (fun i p => ∑ k, ∑ q, f i p k q) = |
| (fun i p => 2 * upperPairSum (f i p)) := by |
| funext i p |
| exact sum_symmetric_pair _ (hsecond i p) (hdiag₂ i p) |
| rw [he, upperPairSum_mul] |
| ring |
|
|
| end Paper170 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
