25 lines
500 B
Plaintext
25 lines
500 B
Plaintext
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import Common.Logic.Basic
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import Mathlib.Data.Set.Intervals.Basic
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namespace Set
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/-! # Common.Set.Intervals
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Additional theorems around intervals.
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-/
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theorem Iio_nat_lt_ssubset {m n : ℕ} (h : m < n)
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: Iio m ⊂ Iio n := by
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rw [ssubset_def]
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apply And.intro
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· unfold Iio
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simp only [setOf_subset_setOf]
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intro x hx
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calc x
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_ < m := hx
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_ < n := h
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· show ¬ ∀ x, x < n → x < m
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simp only [not_forall, not_lt, exists_prop]
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exact ⟨m, h, by simp⟩
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end Set
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