18 lines
511 B
Text
18 lines
511 B
Text
abbrev HasInfinity (α : Type) [LT α] := ∃(inf : α), ∀n, n < inf
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theorem lt_irrefl_imp_no_infinity
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(α : Type) [LT α] (irrefl : ∀(n : α), ¬n < n) : ¬HasInfinity α := by
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intro h
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have ⟨inf, h⟩ := h
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have hlt : inf < inf := h inf
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exact irrefl inf hlt
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theorem no_nat_infinity : ¬HasInfinity Nat :=
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lt_irrefl_imp_no_infinity Nat Nat.lt_irrefl
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theorem no_int_infinity : ¬HasInfinity Nat :=
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lt_irrefl_imp_no_infinity Nat Nat.lt_irrefl
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-- it's… not difficult to see the pattern
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