Add additional proofs to Apostol, Chapter 1.11.

Bookshelf/Apostol/Chapter_1_11.lean

@@ -1,4 +1,7 @@
+import Mathlib.Algebra.BigOperators.Basic
 import Mathlib.Data.Real.Basic
+import Mathlib.Data.Finset.Basic
+import Mathlib.Tactic.LibrarySearch
 
 /-! # Apostol.Chapter_1_11 -/
 
@@ -73,31 +76,77 @@ theorem exercise_4c (x y : ℝ)
       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 4d
+namespace Hermite
 
-`⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2⌋`
+/--
+Constructs a partition of `[0, 1)` that looks as follows:
+```
+[0, 1/n), [1/n, 2/n), ..., [(n-1)/n, 1)
+```
 -/
-theorem exercise_4d (x : ℝ)
+def partition (n : ℕ) (i : ℕ) : Set ℝ := Set.Ico (i / n) ((i + 1) / n)
-  : ⌊2 * x⌋ = ⌊x⌋ + ⌊x + 1/2⌋ := by
+
+/--
+The indexed union of the family of sets of a `partition` is equal to `[0, 1)`.
+-/
+theorem partition_eq_Ico_zero_one
+  : (⋃ i ∈ Finset.range n, partition n i) = Set.Ico 0 1 := by
   sorry
 
-/-- ### Exercise 4e
+/--
-
+The fractional portion of any real number is always in `[0, 1)`.
-`⌊3x⌋ = ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋`
 -/
-theorem exercise_4e (x : ℝ)
+theorem fract_mem_Ico_zero_one (x : ℝ)
-  : ⌊3 * x⌋ = ⌊x⌋ + ⌊x + 1/3⌋ + ⌊x + 2/3⌋ := by
+  : Int.fract x ∈ Set.Ico 0 1 := ⟨Int.fract_nonneg x, Int.fract_lt_one x⟩
+
+/--
+The fractional portion of any real number always exists in some member of the
+indexed family of sets formed by any `partition`.
+-/
+theorem fract_mem_partition (r : ℝ) (hr : r ∈ Set.Ico 0 1)
+  : ∀ n : ℕ, ∃ j : ℕ, r ∈ Set.Ico ↑(j / n) ↑((j + 1) / n) := by
   sorry
 
+end Hermite
+
 /-- ### 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 : ℕ) (x : ℝ)
-  : True := by
+  : ⌊n * x⌋ = Finset.sum (Finset.range n) (fun i => ⌊x + i/n⌋) := by
+  let r := Int.fract x
+  have hx : x = ⌊x⌋ + r := Eq.symm (add_eq_of_eq_sub' rfl)
   sorry
 
+/-- ### Exercise 4d
+
+`⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2⌋`
+-/
+theorem exercise_4d (x : ℝ)
+  : ⌊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 : ℝ)
+  : ⌊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

Bookshelf/Apostol/Chapter_1_11.tex

@@ -1,6 +1,7 @@
 \documentclass{article}
 \usepackage{amsmath}
 \usepackage[shortlabels]{enumitem}
+\usepackage{soul, xcolor}
 
 \input{../../preamble}
 
@@ -88,10 +89,94 @@ The formulas in Exercises 4(d) and 4(e) suggest a generalization for
   $\floor{nx}$.
 State and prove such a generalization.
 
+\vspace{6pt}
+\noindent
+\hl{Note}: The stated generalization is known as "Hermite's Identity."
+
 \begin{proof}
 
   \link{exercise\_5}
 
+  \vspace{6pt}
+  \hrule
+  \vspace{6pt}
+
+  We prove that for all natural numbers $n$ and real numbers $x$, the following
+    identity holds:
+    \begin{equation}
+      \label{sec:exercise-5-eq1}
+      \tag{5.1}
+      \floor{nx} = \sum_{i=0}^{n-1} \floor{x + \frac{i}{n}}
+    \end{equation}
+  By definition of the floor function, $x = \floor{x} + r$ for some
+    $r \in \ico{0}{1}$.
+  Define $S$ as the partition of non-overlapping subintervals
+    $$\ico{0}{\frac{1}{n}}, \ico{\frac{1}{n}}{\frac{2}{n}}, \ldots,
+      \ico{\frac{n-1}{n}}{1}.$$
+  By construction, $\cup\; S = \ico{0}{1}$.
+  Therefore there exists some $j \in \mathbb{N}$ such that
+    \begin{equation}
+      \label{sec:exercise-5-eq2}
+      \tag{5.2}
+      r \in \ico{\frac{j}{n}}{\frac{j+1}{n}}.
+    \end{equation}
+  With these definitions established, we now show the left- and right-hand sides
+    of \eqref{sec:exercise-5-eq1} evaluate to the same number.
+
+  \paragraph{Left-Hand Side}%
+
+    Consider the left-hande side of identity \eqref{sec:exercise-5-eq1}
+    By \eqref{sec:exercise-5-eq2}, $nr \in \ico{j}{j + 1}$.
+    Therefore $\floor{nr} = j$.
+    Thus
+      \begin{align*}
+        \floor{nx}
+          & = \floor{n(\floor{x} + r)} \nonumber \\
+          & = \floor{n\floor{x} + nr} \nonumber \\
+          & = \floor{n\floor{x}} + \floor{nr}. \nonumber
+            & \eqref{sub:exercise-4a} \\
+          & = \floor{n\floor{x}} + j \nonumber \\
+          & = n\floor{x} + j. \label{sec:exercise-5-eq3} \tag{5.3}
+      \end{align*}
+
+  \paragraph{Right-Hand Side}%
+
+    Now consider the right-hand side of identity \eqref{sec:exercise-5-eq1}.
+    We note each summand, by construction, is the floor of $x$ added to a
+      nonnegative number less than one.
+    Therefore each summand contributes either $\floor{x}$ or $\floor{x} + 1$ to
+      the total.
+    Letting $z$ denote the number of summands that contribute $\floor{x} + 1$,
+      we have
+      \begin{equation}
+        \label{sec:exercise-5-eq4}
+        \tag{5.4}
+        \sum_{i=0}^{n-1} \floor{x + \frac{i}{n}} = n\floor{x} + z.
+      \end{equation}
+    The value of $z$ corresponds to the number of indices $i$ that satisfy
+      $$\frac{i}{n} \geq 1 - r.$$
+    By \eqref{sec:exercise-5-eq2}, it follows
+      \begin{align*}
+        1 - r
+          & \in \ioc{1 - \frac{j+1}{n}}{1-\frac{j}{n}} \\
+          & = \ioc{\frac{n - j - 1}{n}}{\frac{n - j}{n}}.
+      \end{align*}
+    Thus we can determine the value of $z$ by instead counting the number of
+      indices $i$ that satisfy $$\frac{i}{n} \geq \frac{n - j}{n}.$$
+    Rearranging terms, we see that $i \geq n - j$ holds for
+      $z = (n - 1) - (n - j) + 1 = j$ of the $n$ summands.
+    Substituting the value of $z$ into \eqref{sec:exercise-5-eq4} yields
+      \begin{equation}
+        \label{sec:exercise-5-eq5}
+        \tag{5.5}
+        \sum_{i=0}^{n-1} \floor{x + \frac{i}{n}} = n\floor{x} + j.
+      \end{equation}
+
+  \paragraph{Conclusion}%
+
+    Since \eqref{sec:exercise-5-eq3} and \eqref{sec:exercise-5-eq5} agree with
+    one another, it follows identity \eqref{sec:exercise-5-eq1} holds.
+
 \end{proof}
 
 \section{Exercise 6}%
@@ -108,7 +193,13 @@ Prove that the number of lattice points in $S$ is equal to the sum
 
 \begin{proof}
 
-  TODO
+  Define $S_i = \mathbb{Z} \cap \ioc{0}{f(i)}$ for all $i \in \mathbb{Z}$.
+  By definition, the set of lattice points of $S$ is given by
+    $$L = \{ (i, j) : i = a, \ldots, b \land j \in S_i \}.$$
+  By construction, it follows $$\sum_{j \in S_i} 1 = \floor{f(i)}.$$
+  Therefore $$\abs{L}
+    = \sum_{i=a}^b \sum_{j \in S_i} 1
+    = \sum_{i=1}^b \floor{f(i)}.$$
 
 \end{proof}
 

preamble.tex

@@ -10,8 +10,13 @@
 
 \hypersetup{colorlinks=true, urlcolor=blue}
 
+\newcommand{\abs}[1]{\left|#1\right|}
 \newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
 \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
+\newcommand{\icc}[2]{\left[#1, #2\right]}
+\newcommand{\ico}[2]{\left[#1, #2\right)}
+\newcommand{\ioc}[2]{\left(#1, #2\right]}
+\newcommand{\ioo}[2]{\left(#1, #2\right)}
 % The first argument refers to a relative path upward from a current file to
 % the root of the workspace (i.e. where this `preamble.tex` file is located).
 \newcommand{\lean}[4]{\href{#1/#2.html\##3}{#4}}