76 lines
1.9 KiB
Plaintext
76 lines
1.9 KiB
Plaintext
import Bookshelf.Real.Int
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import Mathlib.Data.Real.Basic
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/-! # Exercises.Apostol.Exercises_1_11 -/
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namespace Exercises.Apostol.Exercises_1_11
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/-! ## Exercise 4
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Prove that the greatest-integer function has the properties indicated.
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-/
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/-- ### Exercise 4a
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`⌊x + n⌋ = ⌊x⌋ + n` for every integer `n`.
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-/
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theorem exercise_4a (x : ℝ) (n : ℤ) : ⌊x + n⌋ = ⌊x⌋ + n := by
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sorry
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/-- ### Exercise 4b
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`⌊-x⌋ = -⌊x⌋` if `x` is an integer.
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`⌊-x⌋ = -⌊x⌋ - 1` otherwise.
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-/
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theorem exercise_4b (x : ℝ)
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: (Real.isInt x → ⌊-x⌋ = -⌊x⌋)
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∨ (¬Real.isInt x → ⌊-x⌋ = -⌊x⌋ - 1) := by
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sorry
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/-- ### Exercise 4c
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`⌊x + y⌋ = ⌊x⌋ + ⌊y⌋` or `⌊x⌋ + ⌊y⌋ + 1`.
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-/
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theorem exercise_4c (x y : ℝ)
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: ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋ ∨ ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋ + 1 := by
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sorry
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/-- ### Exercise 4d
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`⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2⌋`
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-/
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theorem exercise_4d (x : ℝ)
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: ⌊2 * x⌋ = ⌊x⌋ + ⌊x + 1/2⌋ := by
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sorry
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/-- ### Exercise 4e
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`⌊3x⌋ = ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋`
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-/
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theorem exercise_4e (x : ℝ)
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: ⌊3 * x⌋ = ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋ := by
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sorry
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/-- ### Exercise 5
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The formulas in Exercises 4(d) and 4(e) suggest a generalization for `⌊nx⌋`.
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State and prove such a generalization.
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-/
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theorem exercise_5 (n : ℕ) (x : ℝ)
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: ⌊n * x⌋ = 10 := by
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sorry
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/-- ### Exercise 7b
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If `a` and `b` are positive integers with no common factor, we have the formula
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`Σ_{n=1}^{b-1} ⌊na / b⌋ = ((a - 1)(b - 1)) / 2`. When `b = 1`, the sum on the
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left is understood to be `0`.
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Derive the result analytically as follows: By changing the index of summation,
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note that `Σ_{n=1}^{b-1} ⌊na / b⌋ = Σ_{n=1}^{b-1} ⌊a(b - n) / b⌋`. Now apply
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Exercises 4(a) and (b) to the bracket on the right.
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-/
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theorem exercise_7b : True := sorry
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end Exercises.Apostol.Exercises_1_11
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