Demonstrate how Lean/LaTeX will co-exist for now.

.gitignore

@@ -1,5 +1,22 @@
 # Lean
-**/build
-**/lake-packages/*
-**/_target
-**/leanpkg.path
+*/build
+*/lake-packages
+*/_target
+*/leanpkg.path
+
+# TeX
+*.aux
+*.cb
+*.cb2
+*.fdb_latexmk
+*.fls
+*.fmt
+*.fot
+*.lof
+*.log
+*.lot
+*.out
+*.pdf
+*.synctex.gz
+*.toc
+.*.lb

common/Common/Sequence/Arithmetic.lean

@@ -1,30 +1,38 @@
+import Mathlib.Data.Real.Basic
 import Mathlib.Tactic.NormNum
 import Mathlib.Tactic.Ring
 
-/--[1]
-A 0th-indexed arithmetic sequence.
+/--
+A `0`th-indexed arithmetic sequence.
 -/
 structure Arithmetic where
-  a₀ : Int
-  Δ : Int
+  a₀ : Real
+  Δ : Real
 
 namespace Arithmetic
 
-/--[1]
+/--
 Returns the value of the `n`th term of an arithmetic sequence.
--/
-def termClosed (seq : Arithmetic) (n : Nat) : Int := seq.a₀ + seq.Δ * n
 
-/--[1]
-Returns the value of the `n`th term of an arithmetic sequence.
+This function calculates the value of this term directly. Keep in mind the
+sequence is `0`th-indexed.
 -/
-def termRecursive : Arithmetic → Nat → Int
+def termClosed (seq : Arithmetic) (n : Nat) : Real :=
+  seq.a₀ + seq.Δ * n
+
+/--
+Returns the value of the `n`th term of an arithmetic sequence.
+
+This function calculates the value of this term recursively. Keep in mind the
+sequence is `0`th-indexed.
+-/
+def termRecursive : Arithmetic → Nat → Real
   | seq,       0 => seq.a₀
   | seq, (n + 1) => seq.Δ + seq.termRecursive n
 
-/--[1]
-The recursive definition and closed definitions of an arithmetic sequence are
-equivalent.
+/--
+The recursive and closed term definitions of an arithmetic sequence agree with
+one another.
 -/
 theorem term_recursive_closed (seq : Arithmetic) (n : Nat)
   : seq.termRecursive n = seq.termClosed n := by
@@ -35,14 +43,33 @@ theorem term_recursive_closed (seq : Arithmetic) (n : Nat)
         = seq.Δ + seq.termRecursive n := rfl
       _ = seq.Δ + seq.termClosed n := by rw [ih]
       _ = seq.Δ + (seq.a₀ + seq.Δ * n) := rfl
-      _ = seq.a₀ + seq.Δ * (n + 1) := by ring
+      _ = seq.a₀ + seq.Δ * (↑n + 1) := by ring
+      _ = seq.a₀ + seq.Δ * ↑(n + 1) := by simp
       _ = termClosed seq (n + 1) := rfl
 
-/--[1]
-Summation of the first `n` terms of an arithmetic sequence.
+/--
+The summation of the first `n + 1` terms of an arithmetic sequence.
+
+This function calculates the sum directly.
 -/
-def sum : Arithmetic → Nat → Int
+noncomputable def sum_closed (seq : Arithmetic) (n : Nat) : Real :=
+  ((n + 1) * (seq.a₀ + seq.termClosed n)) / 2
+
+/--
+The summation of the first `n + 1` terms of an arithmetic sequence.
+
+This function calculates the sum recursively.
+-/
+def sum_recursive : Arithmetic → Nat → Real
   |   _,       0 => 0
-  | seq, (n + 1) => seq.termClosed n + seq.sum n
+  | seq, (n + 1) => seq.termClosed (n + 1) + seq.sum_recursive n
+
+/--
+The recursive and closed definitions of the sum of an arithmetic sequence agree
+with one another.
+-/
+theorem sum_recursive_closed (seq : Arithmetic) (n : Nat)
+  : sum_recursive seq n = sum_closed seq n :=
+  sorry
 
 end Arithmetic

common/Common/Sequence/Arithmetic.tex

@@ -0,0 +1,22 @@
+\documentclass{article}
+\usepackage{amsfonts, amsthm}
+
+\newtheorem{theorem}{Theorem}
+
+\begin{document}
+
+\begin{theorem}[Sum of Arithmetic Series]
+
+Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
+Then for some $n \in \mathbb{N}$,
+$$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
+
+\end{theorem}
+
+\begin{proof}
+
+Common.Sequence.Arithmetic.sum\_recursive\_closed
+
+\end{proof}
+
+\end{document}

common/Common/Sequence/Geometric.lean

@@ -1,30 +1,38 @@
+import Mathlib.Data.Real.Basic
 import Mathlib.Tactic.NormNum
 import Mathlib.Tactic.Ring
 
-/--[1]
-A 0th-indexed geometric sequence.
+/--
+A `0th`-indexed geometric sequence.
 -/
 structure Geometric where
-  a₀ : Int
-  r : Int
+  a₀ : Real
+  r : Real
 
 namespace Geometric
 
-/--[1]
-The value of the `n`th term of an geometric sequence.
--/
-def termClosed (seq : Geometric) (n : Nat) : Int := seq.a₀ * seq.r ^ n
+/--
+Returns the value of the `n`th term of a geometric sequence.
 
-/--[1]
-The value of the `n`th term of an geometric sequence.
+This function calculates the value of this term directly. Keep in mind the
+sequence is `0`th-indexed.
 -/
-def termRecursive : Geometric → Nat → Int
+def termClosed (seq : Geometric) (n : Nat) : Real :=
+  seq.a₀ * seq.r ^ n
+
+/--
+Returns the value of the `n`th term of a geometric sequence.
+
+This function calculates the value of this term recursively. Keep in mind the
+sequence is `0`th-indexed.
+-/
+def termRecursive : Geometric → Nat → Real
   | seq,       0 => seq.a₀
   | seq, (n + 1) => seq.r * (seq.termRecursive n)
 
-/--[1]
-The recursive definition and closed definitions of a geometric sequence are
-equivalent.
+/--
+The recursive and closed term definitions of a geometric sequence agree with
+one another.
 -/
 theorem term_recursive_closed (seq : Geometric) (n : Nat)
   : seq.termRecursive n = seq.termClosed n := by
@@ -38,11 +46,30 @@ theorem term_recursive_closed (seq : Geometric) (n : Nat)
       _ = seq.a₀ * seq.r ^ (n + 1) := by ring
       _ = seq.termClosed (n + 1) := rfl
 
-/--[1]
-Summation of the first `n` terms of a geometric sequence.
+/--
+The summation of the first `n + 1` terms of a geometric sequence.
+
+This function calculates the sum directly.
 -/
-def sum : Geometric → Nat → Int
+noncomputable def sum_closed_ratio_neq_one (seq : Geometric) (n : Nat)
+  : seq.r ≠ 1 → Real :=
+  fun _ => (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r)
+
+/--
+The summation of the first `n + 1` terms of a geometric sequence.
+
+This function calculates the sum recursively.
+-/
+def sum_recursive : Geometric → Nat → Real
   |   _,       0 => 0
-  | seq, (n + 1) => seq.termClosed n + seq.sum n
+  | seq, (n + 1) => seq.termClosed n + seq.sum_recursive n
+
+/--
+The recursive and closed definitions of the sum of an arithmetic sequence agree
+with one another.
+-/
+theorem sum_recursive_closed (seq : Geometric) (n : Nat) (p : seq.r ≠ 1)
+  : sum_recursive seq n = sum_closed_ratio_neq_one seq n p :=
+  sorry
 
 end Geometric

common/Common/Sequence/Geometric.tex

@@ -0,0 +1,22 @@
+\documentclass{article}
+\usepackage{amsfonts, amsthm}
+
+\newtheorem{theorem}{Theorem}
+
+\begin{document}
+
+\begin{theorem}[Sum of Geometric Series]
+
+Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
+Then for some $n \in \mathbb{N}$,
+$$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
+
+\end{theorem}
+
+\begin{proof}
+
+Common.Sequence.Geometric.sum\_recursive\_closed.
+
+\end{proof}
+
+\end{document}

common/Common/Tuple.lean

@@ -69,7 +69,8 @@ theorem eq_iff_snoc {t₁ t₂ : Tuple α n}
 /--
 Implements decidable equality for `Tuple α m`, provided `a` has decidable equality. 
 -/
-protected def hasDecEq [DecidableEq α] (t₁ t₂ : Tuple α n) : Decidable (Eq t₁ t₂) :=
+protected def hasDecEq [DecidableEq α] (t₁ t₂ : Tuple α n)
+  : Decidable (Eq t₁ t₂) :=
   match t₁, t₂ with
   | t[], t[] => isTrue eq_nil
   | snoc as a, snoc bs b =>