import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring

/--[1]
A 0th-indexed geometric sequence.
-/
structure Geometric where
  a₀ : Int
  r : Int

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.
-/
theorem term_recursive_closed (seq : Geometric) (n : Nat)
        : seq.termRecursive n = seq.termClosed n := by
  induction n with
  | zero => unfold termClosed termRecursive; norm_num
  | succ 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

/--[1]
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