76 lines
2.1 KiB
Plaintext
76 lines
2.1 KiB
Plaintext
import Mathlib.Data.Real.Basic
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import Mathlib.Tactic.NormNum
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import Mathlib.Tactic.Ring
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/--
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A `0`th-indexed arithmetic sequence.
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-/
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structure Arithmetic where
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a₀ : Real
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Δ : Real
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namespace Arithmetic
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/--
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Returns the value of the `n`th term of an arithmetic sequence.
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This function calculates the value of this term directly. Keep in mind the
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sequence is `0`th-indexed.
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-/
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def termClosed (seq : Arithmetic) (n : Nat) : Real :=
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seq.a₀ + seq.Δ * n
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/--
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Returns the value of the `n`th term of an arithmetic sequence.
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This function calculates the value of this term recursively. Keep in mind the
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sequence is `0`th-indexed.
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-/
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def termRecursive : Arithmetic → Nat → Real
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| seq, 0 => seq.a₀
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| seq, (n + 1) => seq.Δ + seq.termRecursive n
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/--
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The recursive and closed term definitions of an arithmetic sequence agree with
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one another.
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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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_ = seq.a₀ + seq.Δ * ↑(n + 1) := by simp
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_ = termClosed seq (n + 1) := rfl
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/--
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The summation of the first `n + 1` terms of an arithmetic sequence.
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This function calculates the sum directly.
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-/
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noncomputable def sum_closed (seq : Arithmetic) (n : Nat) : Real :=
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((n + 1) * (seq.a₀ + seq.termClosed n)) / 2
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/--
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The summation of the first `n + 1` terms of an arithmetic sequence.
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This function calculates the sum recursively.
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-/
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def sum_recursive : Arithmetic → Nat → Real
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| _, 0 => 0
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| seq, (n + 1) => seq.termClosed (n + 1) + seq.sum_recursive n
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/--
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The recursive and closed definitions of the sum of an arithmetic sequence agree
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with one another.
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-/
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theorem sum_recursive_closed (seq : Arithmetic) (n : Nat)
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: sum_recursive seq n = sum_closed seq n :=
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sorry
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end Arithmetic
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