56 lines
1.4 KiB
Plaintext
56 lines
1.4 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 geometric sequence.
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-/
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structure Geometric where
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a₀ : Int
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r : Int
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namespace Geometric
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/--[1]
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The value of the `n`th term of an geometric sequence.
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-/
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def termClosed (seq : Geometric) (n : Nat) : Int := seq.a₀ * seq.r ^ n
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/--[1]
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The value of the `n`th term of an geometric sequence.
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-/
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def termRecursive : Geometric → Nat → Int
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| seq, 0 => seq.a₀
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| seq, (n + 1) => seq.r * (seq.termRecursive n)
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/--[1]
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The recursive definition and closed definitions of a geometric sequence are
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equivalent.
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-/
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theorem term_recursive_closed (seq : Geometric) (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 termClosed termRecursive; norm_num)
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(fun n ih => calc
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seq.termRecursive (n + 1)
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= seq.r * (seq.termRecursive n) := rfl
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_ = seq.r * (seq.termClosed n) := by rw [ih]
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_ = seq.r * (seq.a₀ * seq.r ^ n) := rfl
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_ = seq.a₀ * seq.r ^ (n + 1) := by ring
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_ = seq.termClosed (n + 1) := rfl)
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/--[1]
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Summation of the first `n` terms of a geometric sequence.
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-/
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def sum : Geometric → 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 Geometric
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