| import OAI.Probability.ContinuumTransition.KacBlocks | |
| namespace OAI | |
| universe u_1 | |
| section | |
| open scoped BigOperators | |
| namespace ContinuumTemperature | |
| open Classical | |
| theorem sum_supported_rows_le {α : Type u_1} [DecidableEq α] (s T : Finset α) (M : ℝ) (hM : 0 ≤ M) | |
| (J : α → α → ℝ) (hrow : ∀ k : s, (∑ l : s, J k l) ≤ M) : | |
| (∑ k : s, ∑ l : s, if (k : α) ∈ T then J k l else 0) ≤ T.card*M := by | |
| have hh : (∑ k : s, ∑ l : s, if (k : α) ∈ T then J k l else 0) ≤ | |
| ∑ k : s, if (k : α) ∈ T then M else 0 := by | |
| apply Finset.sum_le_sum | |
| intro k _ | |
| split_ifs with hk | |
| · simpa only [ite_eq_left hk] using hrow k | |
| · simp only [Finset.sum_const_zero, le_refl] | |
| apply hh.trans | |
| rw [Finset.sum_coe_sort s (fun k : α => if k ∈ T then M else 0), ← Finset.sum_filter] | |
| simp only [Finset.sum_const, nsmul_eq_mul] | |
| have hs : s.filter (fun k => k ∈ T) ⊆ T := fun _ hk => (Finset.mem_filter.mp hk).2 | |
| exact mul_le_mul_of_nonneg_right (by exact_mod_cast Finset.card_le_card hs) hM | |
| end ContinuumTemperature | |
| end | |
| end OAI | |