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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | 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
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