Add initial arithmetic/geometric sequence/series definitions/theorems.

Bookshelf.lean

@@ -1 +1,2 @@
-def hello := "world"
+import Bookshelf.Sequence.Arithmetic
+import Bookshelf.Sequence.Geometric

Bookshelf/Sequence/Arithmetic.lean

@@ -0,0 +1,66 @@
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Ring
+
+/--
+A 0th-indexed arithmetic sequence.
+-/
+structure Arithmetic where
+  a₀ : Int
+  Δ : Int
+
+namespace Arithmetic
+
+/--
+Returns the value of the `n`th term of an arithmetic sequence.
+-/
+def termClosed (seq : Arithmetic) (n : Nat) : Int := seq.a₀ + seq.Δ * n
+
+/--
+Returns the value of the `n`th term of an arithmetic sequence.
+-/
+def termRecursive : Arithmetic → Nat → Int
+  | seq,       0 => seq.a₀
+  | seq, (n + 1) => seq.Δ + seq.termRecursive n
+
+/--
+The recursive definition and closed definitions of an arithmetic sequence are
+equivalent.
+-/
+theorem term_recursive_closed (seq : Arithmetic) (n : Nat)
+        : seq.termRecursive n = seq.termClosed n :=
+  Nat.recOn
+    n
+    (by unfold termRecursive termClosed; norm_num)
+    (fun n ih => calc
+      termRecursive seq (Nat.succ n)
+          = 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
+        _ = termClosed seq (n + 1) := rfl)
+
+/--
+Summation of the first `n` terms of an arithmetic sequence.
+-/
+def sum : Arithmetic → Nat → Int
+  |   _,       0 => 0
+  | seq, (n + 1) => seq.termClosed n + seq.sum n
+
+/--
+The closed formula of the summation of the first `n` terms of an arithmetic
+series.
+--/
+theorem sum_closed_formula (seq : Arithmetic) (n : Nat)
+        : seq.sum n = (n / 2) * (seq.a₀ + seq.termClosed (n - 1)) :=
+  Nat.recOn
+    n
+    (by unfold sum termClosed; norm_num)
+    (fun n ih => calc
+      sum seq n.succ
+          = seq.termClosed n + seq.sum n := rfl
+        _ = seq.termClosed n + (n / 2 * (seq.a₀ + seq.termClosed (n - 1))) := by rw [ih]
+        _ = seq.a₀ + seq.Δ * n + (n / 2 * (seq.a₀ + (seq.a₀ + seq.Δ * ↑(n - 1)))) := rfl
+        -- TODO: To continue, need to find how to deal with division.
+        _ = ↑(n + 1) / 2 * (seq.a₀ + seq.termClosed n) := by sorry)
+
+end Arithmetic

Bookshelf/Sequence/Geometric.lean

@@ -0,0 +1,49 @@
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Ring
+
+/--
+A 0th-indexed geometric sequence.
+-/
+structure Geometric where
+  a₀ : Int
+  r : Int
+
+namespace Geometric
+
+/--
+The value of the `n`th term of an geometric sequence.
+-/
+def termClosed (seq : Geometric) (n : Nat) : Int := seq.a₀ * seq.r ^ n
+
+/--
+The value of the `n`th term of an geometric sequence.
+-/
+def termRecursive : Geometric → Nat → Int
+  | seq,       0 => seq.a₀
+  | seq, (n + 1) => seq.r * (seq.termRecursive n)
+
+/--
+The recursive definition and closed definitions of a geometric sequence are
+equivalent.
+-/
+theorem term_recursive_closed (seq : Geometric) (n : Nat)
+        : seq.termRecursive n = seq.termClosed n :=
+  Nat.recOn
+    n
+    (by unfold termClosed termRecursive; norm_num)
+    (fun n ih => calc
+      seq.termRecursive (n + 1)
+          = seq.r * (seq.termRecursive n) := rfl
+        _ = seq.r * (seq.termClosed n) := by rw [ih]
+        _ = seq.r * (seq.a₀ * seq.r ^ n) := rfl
+        _ = seq.a₀ * seq.r ^ (n + 1) := by ring
+        _ = seq.termClosed (n + 1) := rfl)
+
+/--
+Summation of the first `n` terms of a geometric sequence.
+-/
+def sum : Geometric → Nat → Int
+  |   _,       0 => 0
+  | seq, (n + 1) => seq.termClosed n + seq.sum n
+
+end Geometric

lake-manifest.json

@@ -0,0 +1,27 @@
+{"version": 4,
+ "packagesDir": "lake-packages",
+ "packages":
+ [{"git":
+   {"url": "https://github.com/leanprover-community/mathlib4.git",
+    "subDir?": null,
+    "rev": "065e9956724008d1059577700fd676eaf25f35d5",
+    "name": "mathlib",
+    "inputRev?": null}},
+  {"git":
+   {"url": "https://github.com/gebner/quote4",
+    "subDir?": null,
+    "rev": "7ac99aa3fac487bec1d5860e751b99c7418298cf",
+    "name": "Qq",
+    "inputRev?": "master"}},
+  {"git":
+   {"url": "https://github.com/JLimperg/aesop",
+    "subDir?": null,
+    "rev": "ba61f7fec6174d8c7d2796457da5a8d0b0da44c6",
+    "name": "aesop",
+    "inputRev?": "master"}},
+  {"git":
+   {"url": "https://github.com/leanprover/std4",
+    "subDir?": null,
+    "rev": "de7e2a79905a3f87cad1ad5bf57045206f9738c7",
+    "name": "std",
+    "inputRev?": "main"}}]}

lakefile.lean

@@ -1,6 +1,9 @@
 import Lake
 open Lake DSL
 
+require mathlib from git
+  "https://github.com/leanprover-community/mathlib4.git"
+
 package «bookshelf» {
   -- add package configuration options here
 }