bookshelf/common/Common/Data/Real/Sequence.lean

import Mathlib.Data.Real.Basic
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring

/--
A `0`th-indexed arithmetic sequence.
-/
structure Arithmetic where
  a₀ : Real
  Δ : Real

namespace Arithmetic

/--
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 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

/--
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
  induction n with
  | zero => unfold termRecursive termClosed; norm_num
  | succ 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
      _ = seq.a₀ + seq.Δ * ↑(n + 1) := by simp
      _ = termClosed seq (n + 1) := rfl

/--
A term is equal to the next in the sequence minus the common difference.
-/
theorem term_closed_sub_succ_delta {seq : Arithmetic}
  : seq.termClosed n = seq.termClosed (n + 1) - seq.Δ :=
  calc
    seq.termClosed n
    _ = seq.a₀ + seq.Δ * n := rfl
    _ = seq.a₀ + seq.Δ * n + seq.Δ - seq.Δ := by rw [add_sub_cancel]
    _ = seq.a₀ + seq.Δ * (↑n + 1) - seq.Δ := by ring_nf
    _ = seq.a₀ + seq.Δ * ↑(n + 1) - seq.Δ := by simp only [Nat.cast_add, Nat.cast_one]
    _ = seq.termClosed (n + 1) - seq.Δ := rfl

/--
The summation of the first `n + 1` terms of an arithmetic sequence.

This function calculates the sum directly.
-/
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
  | seq,       0 => seq.a₀
  | seq, (n + 1) => seq.termClosed (n + 1) + seq.sum_recursive n

/--
Simplify a summation of terms found in the proof of `sum_recursive_closed`.
-/
private lemma sub_delta_summand_eq_two_mul_a₀ {seq : Arithmetic}
  : seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ = 2 * seq.a₀ :=
  calc
    seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ
    _ = seq.a₀ + (seq.a₀ + seq.Δ * ↑(n + 1)) - (n + 1) * seq.Δ := rfl
    _ = seq.a₀ + seq.a₀ + seq.Δ * ↑(n + 1) - (n + 1) * seq.Δ := by rw [←add_assoc]
    _ = seq.a₀ + seq.a₀ + seq.Δ * (n + 1) - (n + 1) * seq.Δ := by simp only [Nat.cast_add, Nat.cast_one]
    _ = 2 * seq.a₀ := by ring_nf

/--
The recursive and closed definitions of the sum of an arithmetic sequence agree
with one another.
-/
theorem sum_recursive_closed (seq : Arithmetic) (n : Nat)
  : seq.sum_recursive n = seq.sum_closed n := by
  induction n with
  | zero =>
    unfold sum_recursive sum_closed termClosed
    norm_num
  | succ n ih =>
    calc
      seq.sum_recursive (n + 1)
      _ = seq.termClosed (n + 1) + seq.sum_recursive n := rfl
      _ = seq.termClosed (n + 1) + seq.sum_closed n := by rw [ih]
      _ = seq.termClosed (n + 1) + ((n + 1) * (seq.a₀ + seq.termClosed n)) / 2 := rfl
      _ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed n + seq.a₀ + seq.termClosed n) / 2 := by ring_nf
      _ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * (seq.termClosed (n + 1) - seq.Δ) + seq.a₀ + (seq.termClosed (n + 1) - seq.Δ)) / 2 := by rw [@term_closed_sub_succ_delta n]
      _ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed (n + 1) + (seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ)) / 2 := by ring_nf
      _ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed (n + 1) + 2 * seq.a₀) / 2 := by rw [sub_delta_summand_eq_two_mul_a₀]
      _ = ((n + 1) + 1) * (seq.a₀ + seq.termClosed (n + 1)) / 2 := by ring_nf
      _ = (↑(n + 1) + 1) * (seq.a₀ + seq.termClosed (n + 1)) / 2 := by simp only [Nat.cast_add, Nat.cast_one]
      _ = seq.sum_closed (n + 1) := rfl

end Arithmetic

/--
A `0th`-indexed geometric sequence.
-/
structure Geometric where
  a₀ : Real
  r : Real

namespace Geometric

/--
Returns the value of the `n`th term of a geometric sequence.

This function calculates the value of this term directly. Keep in mind the
sequence is `0`th-indexed.
-/
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)

/--
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
  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

/--
The summation of the first `n + 1` terms of a geometric sequence.

This function calculates the sum directly.
-/
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
  | seq,       0 => seq.a₀
  | seq, (n + 1) => seq.termClosed (n + 1) + seq.sum_recursive n

/--
The recursive and closed definitions of the sum of a geometric 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 := by
  have h : 1 - seq.r ≠ 0 := by
    intro h
    rw [sub_eq_iff_eq_add, zero_add] at h
    exact False.elim (p (Eq.symm h))
  induction n with
  | zero =>
    unfold sum_recursive sum_closed_ratio_neq_one
    simp
    rw [mul_div_assoc, div_self h, mul_one] 
  | succ n ih =>
    calc
      sum_recursive seq (n + 1)
      _ = seq.termClosed (n + 1) + seq.sum_recursive n := rfl
      _ = seq.termClosed (n + 1) + sum_closed_ratio_neq_one seq n p := by rw [ih]
      _ = seq.a₀ * seq.r ^ (n + 1) + (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r) := rfl
      _ = seq.a₀ * seq.r ^ (n + 1) * (1 - seq.r) / (1 - seq.r) + (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r) := by rw [mul_div_cancel _ h]
      _ = (seq.a₀ * (1 - seq.r ^ (n + 1 + 1))) / (1 - seq.r) := by ring_nf
      _ = sum_closed_ratio_neq_one seq (n + 1) p := rfl

end Geometric