bookshelf/Common/Real/Sequence/Arithmetic.lean

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import Mathlib.Data.Real.Basic
/-! # Common.Real.Sequence.Arithmetic
A characterization of an arithmetic sequence, i.e. a sequence with a common
difference between each term.
-/
namespace Real
/--
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 Real.Arithmetic