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9dbc64e | 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 | import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Basic.Real.Basic
namespace OAI
namespace DepthThreeLowerBound
open scoped BigOperators
def trueCount {N : ℕ} (z : Fin N → Bool) : ℕ :=
(Finset.univ.filter fun i => z i = true).card
theorem trueCount_le {N : ℕ} (z : Fin N → Bool) : trueCount z ≤ N := by
simpa only [trueCount, Finset.card_univ, Fintype.card_fin] using
(Finset.card_filter_le (s := (Finset.univ : Finset (Fin N)))
(p := fun i => z i = true))
theorem trueCount_cast {N : ℕ} (z : Fin N → Bool) :
(trueCount z : ℝ) = ∑ i, if z i = true then (1 : ℝ) else 0 := by
exact Finset.natCast_card_filter (fun i => z i = true) Finset.univ
end DepthThreeLowerBound
end OAI
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