bookshelf/Bookshelf/Apostol/Chapter_1_11.lean

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import Common.Real.Floor
import Common.Set.Basic
import Mathlib.Data.Real.Basic
/-! # Apostol.Chapter_1_11 -/
namespace Apostol.Chapter_1_11
open BigOperators
/-! ## Exercise 4
Prove that the greatest-integer function has the properties indicated.
-/
/-- ### Exercise 4a
`⌊x + n⌋ = ⌊x⌋ + n` for every integer `n`.
-/
theorem exercise_4a (x : ) (n : ) : ⌊x + n⌋ = ⌊x⌋ + n :=
Int.floor_add_int x n
/-- ### Exercise 4b.1
`⌊-x⌋ = -⌊x⌋` if `x` is an integer.
-/
theorem exercise_4b_1 (x : ) : ⌊-x⌋ = -⌊x⌋ := by
simp only [Int.floor_int, id_eq]
/-- ### Exercise 4b.2
`⌊-x⌋ = -⌊x⌋ - 1` otherwise.
-/
theorem exercise_4b_2 (x : ) (h : ∃ n : , x ∈ Set.Ioo ↑n (↑n + (1 : )))
: ⌊-x⌋ = -⌊x⌋ - 1 := by
rw [Int.floor_neg]
suffices ⌈x⌉ = ⌊x⌋ + 1 by
have := congrArg (HMul.hMul (-1)) this
simp only [neg_mul, one_mul, neg_add_rev, add_comm] at this
exact this
have ⟨n, hn⟩ := h
have hn' : x ∈ Set.Ico ↑n (↑n + (1 : )) :=
Set.mem_of_subset_of_mem Set.Ioo_subset_Ico_self hn
rw [Int.ceil_eq_iff, Int.floor_eq_on_Ico n x hn']
simp only [Int.cast_add, Int.cast_one, add_sub_cancel]
apply And.intro
· exact (Set.mem_Ioo.mp hn).left
· exact le_of_lt (Set.mem_Ico.mp hn').right
/-- ### Exercise 4c
`⌊x + y⌋ = ⌊x⌋ + ⌊y⌋` or `⌊x⌋ + ⌊y⌋ + 1`.
-/
theorem exercise_4c (x y : )
: ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋ ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋ + 1 := by
have hx : x = Int.floor x + Int.fract x := Eq.symm (add_eq_of_eq_sub' rfl)
have hy : y = Int.floor y + Int.fract y := Eq.symm (add_eq_of_eq_sub' rfl)
by_cases Int.fract x + Int.fract y < 1
· refine Or.inl ?_
rw [Int.floor_eq_iff]
simp only [Int.cast_add]
apply And.intro
· exact add_le_add (Int.floor_le x) (Int.floor_le y)
· conv => lhs; rw [hx, hy, add_add_add_comm]; arg 1; rw [add_comm]
rwa [add_comm, ← add_assoc, ← sub_lt_iff_lt_add', ← sub_sub, add_sub_cancel, add_sub_cancel]
· refine Or.inr ?_
rw [Int.floor_eq_iff]
simp only [Int.cast_add, Int.cast_one]
have h := le_of_not_lt h
apply And.intro
· conv => lhs; rw [← add_rotate]
conv => rhs; rw [hx, hy, add_add_add_comm]; arg 1; rw [add_comm]
rwa [← sub_le_iff_le_add', ← sub_sub, add_sub_cancel, add_sub_cancel]
· conv => lhs; rw [hx, hy, add_add_add_comm]; arg 1; rw [add_comm]
conv => lhs; rw [add_comm, ← add_assoc]
conv => rhs; rw [add_assoc]
rw [← sub_lt_iff_lt_add', ← sub_sub, add_sub_cancel, add_sub_cancel]
exact add_lt_add (Int.fract_lt_one x) (Int.fract_lt_one y)
/-- ### Exercise 5
The formulas in Exercises 4(d) and 4(e) suggest a generalization for `⌊nx⌋`.
State and prove such a generalization.
-/
theorem exercise_5 (n : ) (x : )
: ⌊n * x⌋ = Finset.sum (Finset.range n) (fun i => ⌊x + i/n⌋) :=
Real.Floor.floor_mul_eq_sum_range_floor_add_index_div n x
/-- ### Exercise 4d
`⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2⌋`
-/
theorem exercise_4d (x : )
: ⌊2 * x⌋ = ⌊x⌋ + ⌊x + 1/2⌋ := by
suffices ⌊x⌋ + ⌊x + 1/2⌋ = Finset.sum (Finset.range 2) (fun i => ⌊x + i/2⌋) by
rw [this]
exact exercise_5 2 x
unfold Finset.sum
simp
rw [add_comm]
/-- ### Exercise 4e
`⌊3x⌋ = ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋`
-/
theorem exercise_4e (x : )
: ⌊3 * x⌋ = ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋ := by
suffices ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋ = Finset.sum (Finset.range 3) (fun i => ⌊x + i/3⌋) by
rw [this]
exact exercise_5 3 x
unfold Finset.sum
simp
conv => rhs; rw [← add_rotate']; arg 2; rw [add_comm]
rw [← add_assoc]
/-- ### Exercise 7b
If `a` and `b` are positive integers with no common factor, we have the formula
`∑_{n=1}^{b-1} ⌊na / b⌋ = ((a - 1)(b - 1)) / 2`. When `b = 1`, the sum on the
left is understood to be `0`.
Derive the result analytically as follows: By changing the index of summation,
note that `Σ_{n=1}^{b-1} ⌊na / b⌋ = Σ_{n=1}^{b-1} ⌊a(b - n) / b⌋`. Now apply
Exercises 4(a) and (b) to the bracket on the right.
-/
theorem exercise_7b (ha : a > 0) (hb : b > 0) (hp : Nat.coprime a b)
: ∑ n in (Finset.range b).filter (· > 0), ⌊n * ((a : ) : ) / b⌋ =
((a - 1) * (b - 1)) / 2 := by
sorry
section
/-- ### Exercise 8
Let `S` be a set of points on the real line. The *characteristic function* of
`S` is, by definition, the function `Χ` such that `Χₛ(x) = 1` for every `x` in
`S`, and `Χₛ(x) = 0` for those `x` not in `S`. Let `f` be a step function which
takes the constant value `cₖ` on the `k`th open subinterval `Iₖ` of some
partition of an interval `[a, b]`. Prove that for each `x` in the union
`I₁ I₂ Iₙ` we have
```
f(x) = ∑_{k=1}^n cₖΧ_{Iₖ}(x).
```
This property is described by saying that every step function is a linear
combination of characteristic functions of intervals.
-/
theorem exercise_8 : True := sorry
end
end Apostol.Chapter_1_11