49 lines
1.3 KiB
Plaintext
49 lines
1.3 KiB
Plaintext
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 := by
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induction n with
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| zero => unfold termRecursive termClosed; norm_num
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| succ 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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end Arithmetic
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