2023-04-09 22:27:34 +00:00
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import Mathlib.Data.PNat.Basic
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import Mathlib.Data.Real.Basic
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#check Archimedean
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namespace Real
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/--
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Every real should be less than or equal to the absolute value of its ceiling.
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-/
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lemma leq_nat_abs_ceil_self (x : ℝ) : x ≤ Int.natAbs ⌈x⌉ := by
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by_cases h : x ≥ 0
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· let k : ℤ := ⌈x⌉
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unfold Int.natAbs
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have k' : k = ⌈x⌉ := rfl
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rw [←k']
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have _ : k ≥ 0 := by -- Hint for match below
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rw [k', ge_iff_le]
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exact Int.ceil_nonneg (ge_iff_le.mp h)
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match k with
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| Int.ofNat m => calc x
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_ ≤ ⌈x⌉ := Int.le_ceil x
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_ = Int.ofNat m := by rw [←k']
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· have h' : ((Int.natAbs ⌈x⌉) : ℝ) ≥ 0 := by simp
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calc x
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_ ≤ 0 := le_of_lt (lt_of_not_le h)
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_ ≤ ↑(Int.natAbs ⌈x⌉) := GE.ge.le h'
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/--
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Theorem 1.29
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For every real `x` there exists a positive integer `n` such that `n > x`.
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-/
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theorem exists_pnat_geq_self (x : ℝ) : ∃ n : ℕ+, ↑n > x := by
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let x' : ℕ+ := ⟨Int.natAbs ⌈x⌉ + 1, by simp⟩
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have h : x < x' := calc x
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_ ≤ Int.natAbs ⌈x⌉ := leq_nat_abs_ceil_self x
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_ < ↑↑(Int.natAbs ⌈x⌉ + 1) := by simp
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_ = x' := rfl
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exact ⟨x', h⟩
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/--
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Theorem 1.30
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If `x > 0` and if `y` is an arbitrary real number, there exists a positive
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integer `n` such that `nx > y`.
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This is known as the *Archimedean Property of the Reals*.
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-/
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theorem pos_imp_exists_pnat_mul_self_geq {x y : ℝ}
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: x > 0 → ∃ n : ℕ+, n * x > y := by
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intro hx
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let ⟨n, p⟩ := exists_pnat_geq_self (y / x)
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have p' := mul_lt_mul_of_pos_right p hx
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rw [div_mul, div_self (show x ≠ 0 from LT.lt.ne' hx), div_one] at p'
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exact ⟨n, p'⟩
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2023-04-09 22:28:55 +00:00
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end Real
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