73 lines
2.1 KiB
Plaintext
73 lines
2.1 KiB
Plaintext
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/-
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# References
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1. Levin, Oscar. Discrete Mathematics: An Open Introduction. 3rd ed., n.d.
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https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
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-/
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import Mathlib.Tactic.NormNum
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import Mathlib.Tactic.Ring
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/--[1]
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A 0th-indexed arithmetic sequence.
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-/
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structure Arithmetic where
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a₀ : Int
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Δ : Int
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namespace Arithmetic
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/--[1]
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Returns the value of the `n`th term of an arithmetic sequence.
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-/
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def termClosed (seq : Arithmetic) (n : Nat) : Int := seq.a₀ + seq.Δ * n
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/--[1]
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Returns the value of the `n`th term of an arithmetic sequence.
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-/
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def termRecursive : Arithmetic → Nat → Int
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| seq, 0 => seq.a₀
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| seq, (n + 1) => seq.Δ + seq.termRecursive n
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/--[1]
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The recursive definition and closed definitions of an arithmetic sequence are
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equivalent.
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-/
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theorem term_recursive_closed (seq : Arithmetic) (n : Nat)
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: seq.termRecursive n = seq.termClosed n :=
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Nat.recOn
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n
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(by unfold termRecursive termClosed; norm_num)
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(fun n ih => calc
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termRecursive seq (Nat.succ n)
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= seq.Δ + seq.termRecursive n := rfl
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_ = seq.Δ + seq.termClosed n := by rw [ih]
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_ = seq.Δ + (seq.a₀ + seq.Δ * n) := rfl
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_ = seq.a₀ + seq.Δ * (n + 1) := by ring
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_ = termClosed seq (n + 1) := rfl)
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/--[1]
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Summation of the first `n` terms of an arithmetic sequence.
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-/
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def sum : Arithmetic → Nat → Int
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| _, 0 => 0
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| seq, (n + 1) => seq.termClosed n + seq.sum n
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/--[1]
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The closed formula of the summation of the first `n` terms of an arithmetic
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series.
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--/
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theorem sum_closed_formula (seq : Arithmetic) (n : Nat)
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: seq.sum n = (n / 2) * (seq.a₀ + seq.termClosed (n - 1)) :=
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Nat.recOn
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n
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(by unfold sum termClosed; norm_num)
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(fun n ih => calc
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sum seq n.succ
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= seq.termClosed n + seq.sum n := rfl
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_ = seq.termClosed n + (n / 2 * (seq.a₀ + seq.termClosed (n - 1))) := by rw [ih]
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_ = seq.a₀ + seq.Δ * n + (n / 2 * (seq.a₀ + (seq.a₀ + seq.Δ * ↑(n - 1)))) := rfl
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-- TODO: To continue, need to find how to deal with division.
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_ = ↑(n + 1) / 2 * (seq.a₀ + seq.termClosed n) := by sorry)
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end Arithmetic
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