98 lines
3.0 KiB
Plaintext
98 lines
3.0 KiB
Plaintext
import Mathlib.Data.Real.Basic
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/-! # Bookshelf.Real.Sequence.Geometric
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A characterization of a geometric sequence, i.e. a sequence with a common ratio
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between each term.
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-/
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namespace Real
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/--
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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 : Real
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r : Real
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namespace Geometric
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/--
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Returns the value of the `n`th term of a geometric 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 : Geometric) (n : Nat) : Real :=
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seq.a₀ * seq.r ^ n
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/--
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Returns the value of the `n`th term of a geometric 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 : Geometric → Nat → Real
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| seq, 0 => seq.a₀
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| seq, (n + 1) => seq.r * (seq.termRecursive n)
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/--
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The recursive and closed term definitions of a geometric sequence agree with
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one another.
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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 := by
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induction n with
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| zero => unfold termClosed termRecursive; norm_num
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| succ 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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/--
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The summation of the first `n + 1` terms of a geometric sequence.
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This function calculates the sum directly.
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-/
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noncomputable def sum_closed_ratio_neq_one (seq : Geometric) (n : Nat)
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: seq.r ≠ 1 → Real :=
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fun _ => (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r)
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/--
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The summation of the first `n + 1` terms of a geometric sequence.
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This function calculates the sum recursively.
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-/
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def sum_recursive : Geometric → Nat → Real
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| seq, 0 => seq.a₀
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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 a geometric sequence agree
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with one another.
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-/
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theorem sum_recursive_closed (seq : Geometric) (n : Nat) (p : seq.r ≠ 1)
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: sum_recursive seq n = sum_closed_ratio_neq_one seq n p := by
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have h : 1 - seq.r ≠ 0 := by
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intro h
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rw [sub_eq_iff_eq_add, zero_add] at h
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exact False.elim (p (Eq.symm h))
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induction n with
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| zero =>
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unfold sum_recursive sum_closed_ratio_neq_one
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simp
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rw [mul_div_assoc, div_self h, mul_one]
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| succ n ih =>
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calc
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sum_recursive seq (n + 1)
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_ = seq.termClosed (n + 1) + seq.sum_recursive n := rfl
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_ = seq.termClosed (n + 1) + sum_closed_ratio_neq_one seq n p := by rw [ih]
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_ = seq.a₀ * seq.r ^ (n + 1) + (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r) := rfl
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_ = seq.a₀ * seq.r ^ (n + 1) * (1 - seq.r) / (1 - seq.r) + (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r) := by rw [mul_div_cancel _ h]
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_ = (seq.a₀ * (1 - seq.r ^ (n + 1 + 1))) / (1 - seq.r) := by ring_nf
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_ = sum_closed_ratio_neq_one seq (n + 1) p := rfl
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end Real.Geometric |