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f205175 | 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 33 | import Mathlib
namespace OAI
section
section
noncomputable section
open Set
open scoped Classical
namespace CriticalPercolation
def integerPairBox (k : ℕ) : Set (Multiplicative (ℤ×ℤ)) :=
{t | |(t.toAdd.1 : ℝ)|≤(k:ℝ) ∧ |(t.toAdd.2 : ℝ)|≤(k:ℝ)}
lemma integerPairBox_finite (k : ℕ) : (integerPairBox k).Finite := by
have hh := ((Set.finite_Icc (-(k:ℤ)) (k:ℤ)).prod
(Set.finite_Icc (-(k:ℤ)) (k:ℤ))).image (Multiplicative.ofAdd : ℤ×ℤ → _)
apply hh.subset
intro t ht
refine ⟨t.toAdd,?_,rfl⟩
have h₁ := abs_le.mp ht.1
have h₂ := abs_le.mp ht.2
exact ⟨⟨by exact_mod_cast h₁.1,by exact_mod_cast h₁.2⟩,
⟨by exact_mod_cast h₂.1,by exact_mod_cast h₂.2⟩⟩
end CriticalPercolation
end
end
end
end OAI
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