Add `Tuple` module for use in mathematical-introduction-logic, chapter 1.

bookshelf/mathematical-introduction-logic/MathematicalIntroductionLogic.lean

@@ -0,0 +1,7 @@
+/-
+# References
+
+1. Enderton, Herbert B. A Mathematical Introduction to Logic. 2nd ed. San Diego:
+   Harcourt/Academic Press, 2001.
+-/
+import MathematicalIntroductionLogic.Chapter0

bookshelf/mathematical-introduction-logic/MathematicalIntroductionLogic/Chapter0.lean

@@ -0,0 +1,6 @@
+/-
+# References
+
+1. Enderton, Herbert B. A Mathematical Introduction to Logic. 2nd ed. San Diego:
+   Harcourt/Academic Press, 2001.
+-/

bookshelf/mathematical-introduction-logic/lakefile.lean

@@ -0,0 +1,9 @@
+import Lake
+open Lake DSL
+
+package «mathematical-introduction-logic»
+
+@[default_target]
+lean_lib «MathematicalIntroductionLogic» {
+  -- add library configuration options here
+}

bookshelf/mathematical-introduction-logic/lean-toolchain

@@ -0,0 +1 @@
+leanprover/lean4:nightly-2023-02-12

common/Bookshelf.lean

@@ -0,0 +1,3 @@
+import Bookshelf.Sequence.Arithmetic
+import Bookshelf.Sequence.Geometric
+import Bookshelf.Tuple

common/Bookshelf/Sequence/Arithmetic.lean

@@ -1,7 +1,14 @@
+/-
+# References
+
+1. Levin, Oscar. Discrete Mathematics: An Open Introduction. 3rd ed., n.d.
+   https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
+-/
+
 import Mathlib.Tactic.NormNum
 import Mathlib.Tactic.Ring
 
-/--
+/--[1]
 A 0th-indexed arithmetic sequence.
 -/
 structure Arithmetic where
@@ -10,19 +17,19 @@ structure Arithmetic where
 
 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.
 -/
 def termRecursive : Arithmetic → Nat → Int
   | 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.
 -/
@@ -39,14 +46,14 @@ theorem term_recursive_closed (seq : Arithmetic) (n : Nat)
         _ = seq.a₀ + seq.Δ * (n + 1) := by ring
         _ = termClosed seq (n + 1) := rfl)
 
-/--
+/--[1]
 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
 
-/--
+/--[1]
 The closed formula of the summation of the first `n` terms of an arithmetic
 series.
 --/

common/Bookshelf/Sequence/Geometric.lean

@@ -1,7 +1,14 @@
+/-
+# References
+
+1. Levin, Oscar. Discrete Mathematics: An Open Introduction. 3rd ed., n.d.
+   https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
+-/
+
 import Mathlib.Tactic.NormNum
 import Mathlib.Tactic.Ring
 
-/--
+/--[1]
 A 0th-indexed geometric sequence.
 -/
 structure Geometric where
@@ -10,19 +17,19 @@ structure Geometric where
 
 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
 
-/--
+/--[1]
 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)
 
-/--
+/--[1]
 The recursive definition and closed definitions of a geometric sequence are
 equivalent.
 -/
@@ -39,7 +46,7 @@ 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.
 -/
 def sum : Geometric → Nat → Int

common/Bookshelf/Tuple.lean

@@ -0,0 +1,59 @@
+/-
+# References
+
+1. Enderton, Herbert B. A Mathematical Introduction to Logic. 2nd ed. San Diego:
+   Harcourt/Academic Press, 2001.
+-/
+
+import Mathlib.Tactic.Ring
+
+universe u
+
+/--[1]
+An n-tuple is defined recursively as:
+
+  ⟨x₁, ..., xₙ₊₁⟩ = ⟨⟨x₁, ..., xₙ⟩, xₙ₊₁⟩
+
+As [1] notes, it is also useful to define ⟨x⟩ = x.
+--/
+inductive Tuple (α : Type u) : Nat → Type u where
+  | nil : Tuple α 0
+  | cons : {n : Nat} → Tuple α n → α → Tuple α (n + 1)
+
+syntax (priority := high) "⟨" term,+ "⟩" : term
+
+macro_rules
+  | `(⟨$x⟩) => `(Tuple.cons Tuple.nil $x)
+  | `(⟨$xs:term,*, $x⟩) => `(Tuple.cons ⟨$xs,*⟩ $x)
+
+namespace Tuple
+
+/--
+Returns the value at the nth-index of the given tuple.
+-/
+def index (t : Tuple α n) (m : Nat) : 1 ≤ m ∧ m ≤ n → α := by
+  intro h
+  cases t
+  · case nil =>
+    have ff : 1 ≤ 0 := Nat.le_trans h.left h.right
+    ring_nf at ff
+    exact False.elim ff
+  . case cons n' init last =>
+    by_cases k : m = n' + 1
+    · exact last
+    · exact index init m (And.intro h.left (by
+        have h₂ : m + 1 ≤ n' + 1 := Nat.lt_of_le_of_ne h.right k
+        norm_num at h₂
+        exact h₂))
+
+-- TODO: Prove the following theorem
+
+theorem eq_by_index (t₁ t₂ : Tuple α n)
+  : (t₁ = t₂) ↔ (∀ i : Nat, 1 ≤ i ∧ i ≤ n → index t₁ i = index t₂ i) := by
+  apply Iff.intro
+  · sorry
+  · sorry
+
+-- TODO: [1] Lemma 0A
+
+end Tuple

common/Sequence.lean

@@ -1,2 +0,0 @@
-import Sequence.Arithmetic
-import Sequence.Geometric

common/lakefile.lean

@@ -8,5 +8,5 @@ package «Common»
 
 @[default_target]
 lean_lib «Common» {
-  roots := #["Sequence"]
+  roots := #["Bookshelf"]
 }