Apostol I 3.
Better organize concepts in `common` and continue adding more to parts of I 3 proofs.finite-set-exercises
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@ -1,3 +1,3 @@
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import Common.Sequence.Arithmetic
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import Common.Sequence.Geometric
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import Common.Data.Real.Set
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import Common.Data.Real.Sequence
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import Common.Tuple
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@ -115,3 +115,92 @@ theorem sum_recursive_closed (seq : Arithmetic) (n : Nat)
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_ = seq.sum_closed (n + 1) := rfl
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end Arithmetic
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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 Geometric
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@ -0,0 +1,35 @@
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\documentclass{article}
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\input{../../../preamble}
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\begin{document}
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\begin{theorem}[Sum of Arithmetic Series]
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Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
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Then for some $n \in \mathbb{N}$,
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$$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
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\end{theorem}
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\begin{proof}
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\href{Sequence.lean}{Common.Data.Real.Sequence.Arithmetic.sum_recursive_closed}
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\end{proof}
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\begin{theorem}[Sum of Geometric Series]
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Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
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Then for some $n \in \mathbb{N}$,
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$$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
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\end{theorem}
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\begin{proof}
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\href{Sequence.lean}{Common.Data.Real.Sequence.Geometric.sum_recursive_closed}
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\end{proof}
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\end{document}
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@ -0,0 +1,12 @@
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import Mathlib.Data.Real.Basic
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namespace Real
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/--
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The Minkowski sum of two sets `s` and `t` is the set
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`s + t = { a + b : a ∈ s, b ∈ t }`.
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-/
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def minkowski_sum (s t : Set ℝ) :=
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{ x | ∃ a ∈ s, ∃ b ∈ t, x = a + b }
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end Real
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@ -1,21 +0,0 @@
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\documentclass{article}
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\input{../../preamble}
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\begin{document}
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\begin{theorem}[Sum of Arithmetic Series]
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Let $(a_i)_{i \geq 0}$ be an arithmetic sequence with common difference $d$.
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Then for some $n \in \mathbb{N}$,
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$$\sum_{i=0}^n a_i = \frac{(n + 1)(a_0 + a_n)}{2}.$$
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\end{theorem}
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\begin{proof}
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\href{Arithmetic.lean}{Common.Sequence.Arithmetic.sum_recursive_closed}
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\end{proof}
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\end{document}
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@ -1,92 +0,0 @@
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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 `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 Geometric
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@ -1,21 +0,0 @@
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\documentclass{article}
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\input{../../preamble}
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\begin{document}
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\begin{theorem}[Sum of Geometric Series]
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Let $(a_i)_{i \geq 0}$ be a geometric sequence with common ratio $r \neq 1$.
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Then for some $n \in \mathbb{N}$,
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$$\sum_{i=0}^n a_i = \frac{a_0(1 - r^{n+1})}{1 - r}.$$
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\end{theorem}
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\begin{proof}
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\href{Geometric.lean}{Common.Sequence.Geometric.sum_recursive_closed}
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\end{proof}
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\end{document}
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@ -1,4 +1,6 @@
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import Common.Data.Real.Set
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import Mathlib.Data.Real.Basic
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import Mathlib.Data.Set.Basic
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import Mathlib.Data.Set.Pointwise.Basic
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import Mathlib.Tactic.LibrarySearch
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@ -7,6 +9,10 @@ import Mathlib.Tactic.LibrarySearch
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namespace Real
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-- ========================================
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-- The least-upper-bound axiom (completeness axiom)
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-- ========================================
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/--
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A property holds for the negation of elements in set `S` if and only if it also
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holds for the elements of the negation of `S`.
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@ -139,6 +145,10 @@ lemma leq_nat_abs_ceil_self (x : ℝ) : x ≤ Int.natAbs ⌈x⌉ := by
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_ ≤ 0 := le_of_lt (lt_of_not_le h)
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_ ≤ ↑(Int.natAbs ⌈x⌉) := GE.ge.le h'
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-- ========================================
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-- The Archimedean property of the real-number system
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-- ========================================
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/--
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Theorem I.29
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@ -200,14 +210,75 @@ theorem forall_pnat_leq_self_leq_frac_imp_eq {x y a : ℝ}
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_ ≤ a + y / ↑↑n := (h n).right
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simp at this
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-- ========================================
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-- Fundamental properties of the supremum and infimum
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-- ========================================
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/--
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Every member of a set `S` is less than or equal to some value `ub` if and only
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if `ub` is an upper bound of `S`.
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-/
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lemma mem_upper_bounds_iff_forall_le {S : Set ℝ}
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: ub ∈ upperBounds S ↔ (∀ x, x ∈ S → x ≤ ub) := by
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apply Iff.intro
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· intro h _ hx
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exact (h hx)
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· exact id
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/--
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If a set `S` has a least upper bound, it follows every member of `S` is less
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than or equal to that value.
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-/
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lemma forall_lub_imp_forall_le {S : Set ℝ}
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: IsLUB S ub → (∀ x, x ∈ S → x ≤ ub) := by
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intro h
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rw [← mem_upper_bounds_iff_forall_le]
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exact h.left
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/--
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Theorem I.32a
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Let `h` be a given positive number and let `S` be a set of real numbers. If `S`
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has a supremum, then for some `x` in `S` we have `x > sup S - h`.
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-/
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theorem arb_close_to_sup (S : Set ℝ) (s h : ℝ) (hp : h > 0)
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: IsLUB S s → ∃ x : S, x > s - h := sorry
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theorem sup_imp_exists_gt_sup_sub_delta (S : Set ℝ) (s h : ℝ) (hp : h > 0)
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: IsLUB S s → ∃ x ∈ S, x > s - h := by
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intro hb
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-- Suppose all members of our set was less than `s - h`. Then `s` couldn't be
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-- the *least* upper bound.
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by_contra nb
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suffices s - h ∈ upperBounds S by
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have h' : h < h := calc h
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_ ≤ 0 := (le_sub_self_iff s).mp (hb.right this)
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_ < h := hp
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simp at h'
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rw [not_exists] at nb
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have nb' : ∀ x ∈ S, x ≤ s - h := by
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intro x hx
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exact le_of_not_gt (not_and.mp (nb x) hx)
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rw [← mem_upper_bounds_iff_forall_le] at nb'
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exact nb'
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/--
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Every member of a set `S` is greater than or equal to some value `lb` if and
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only if `lb` is a lower bound of `S`.
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-/
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lemma mem_lower_bounds_iff_forall_ge {S : Set ℝ}
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: lb ∈ lowerBounds S ↔ (∀ x ∈ S, x ≥ lb) := by
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apply Iff.intro
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· intro h _ hx
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exact (h hx)
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· exact id
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/--
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If a set `S` has a greatest lower bound, it follows every member of `S` is
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greater than or equal to that value.
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-/
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lemma forall_glb_imp_forall_ge {S : Set ℝ}
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: IsGLB S lb → (∀ x ∈ S, x ≥ lb) := by
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intro h
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rw [← mem_lower_bounds_iff_forall_ge]
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exact h.left
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/--
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Theorem I.32b
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@ -215,7 +286,97 @@ Theorem I.32b
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Let `h` be a given positive number and let `S` be a set of real numbers. If `S`
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has an infimum, then for some `x` in `S` we have `x < inf S + h`.
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-/
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theorem arb_close_to_inf (S : Set ℝ) (s h : ℝ) (hp : h > 0)
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: IsGLB S s → ∃ x : S, x < s + h := sorry
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theorem inf_imp_exists_lt_inf_add_delta (S : Set ℝ) (s h : ℝ) (hp : h > 0)
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: IsGLB S s → ∃ x ∈ S, x < s + h := by
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intro hb
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-- Suppose all members of our set was greater than `s + h`. Then `s` couldn't
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-- be the *greatest* lower bound.
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by_contra nb
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suffices s + h ∈ lowerBounds S by
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have h' : h < h := calc h
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_ ≤ 0 := (add_le_iff_nonpos_right s).mp (hb.right this)
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_ < h := hp
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simp at h'
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rw [not_exists] at nb
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have nb' : ∀ x ∈ S, x ≥ s + h := by
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intro x hx
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exact le_of_not_gt (not_and.mp (nb x) hx)
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rw [← mem_lower_bounds_iff_forall_ge] at nb'
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exact nb'
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/--
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Theorem I.33a (Additive Property)
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Given nonempty subsets `A` and `B` of `ℝ`, let `C` denote the set
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`C = {a + b : a ∈ A, b ∈ B}`. If each of `A` and `B` has a supremum, then `C`
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has a supremum, and `sup C = sup A + sup B`.
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-/
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theorem sup_minkowski_sum_eq_sup_add_sup (A B : Set ℝ) (a b : ℝ)
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(hA : A.Nonempty) (hB : B.Nonempty)
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(ha : IsLUB A a) (hb : IsLUB B b)
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: IsLUB (Real.minkowski_sum A B) (a + b) := by
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let C := Real.minkowski_sum A B
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-- First we show `a + b` is an upper bound of `C`.
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have hub : a + b ∈ upperBounds C := by
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rw [mem_upper_bounds_iff_forall_le]
|
||||
intro x hx
|
||||
have ⟨a', ⟨ha', ⟨b', ⟨hb', hxab⟩⟩⟩⟩: ∃ a ∈ A, ∃ b ∈ B, x = a + b := hx
|
||||
have hs₁ : a' ≤ a := (forall_lub_imp_forall_le ha) a' ha'
|
||||
have hs₂ : b' ≤ b := (forall_lub_imp_forall_le hb) b' hb'
|
||||
exact calc x
|
||||
_ = a' + b' := hxab
|
||||
_ ≤ a + b := add_le_add hs₁ hs₂
|
||||
-- Now we show `a + b` is the *least* upper bound of `C`. If it wasn't, then
|
||||
-- either `¬IsLUB A a` or `¬IsLUB B b`, a contradiction.
|
||||
by_contra nub
|
||||
have h : ¬IsLUB A a ∨ ¬IsLUB B b := by
|
||||
sorry
|
||||
cases h with
|
||||
| inl na => exact absurd ha na
|
||||
| inr nb => exact absurd hb nb
|
||||
|
||||
/--
|
||||
Theorem I.33b (Additive Property)
|
||||
|
||||
Given nonempty subsets `A` and `B` of `ℝ`, let `C` denote the set
|
||||
`C = {a + b : a ∈ A, b ∈ B}`. If each of `A` and `B` has an infimum, then `C`
|
||||
has an infimum, and `inf C = inf A + inf B`.
|
||||
-/
|
||||
theorem inf_minkowski_sum_eq_inf_add_inf (A B : Set ℝ)
|
||||
(hA : A.Nonempty) (hB : B.Nonempty)
|
||||
(ha : IsGLB A a) (hb : IsGLB B b)
|
||||
: IsGLB (Real.minkowski_sum A B) (a + b) := by
|
||||
let C := Real.minkowski_sum A B
|
||||
-- First we show `a + b` is a lower bound of `C`.
|
||||
have hub : a + b ∈ lowerBounds C := by
|
||||
rw [mem_lower_bounds_iff_forall_ge]
|
||||
intro x hx
|
||||
have ⟨a', ⟨ha', ⟨b', ⟨hb', hxab⟩⟩⟩⟩: ∃ a ∈ A, ∃ b ∈ B, x = a + b := hx
|
||||
have hs₁ : a' ≥ a := (forall_glb_imp_forall_ge ha) a' ha'
|
||||
have hs₂ : b' ≥ b := (forall_glb_imp_forall_ge hb) b' hb'
|
||||
exact calc x
|
||||
_ = a' + b' := hxab
|
||||
_ ≥ a + b := add_le_add hs₁ hs₂
|
||||
-- Now we show `a + b` is the *greatest* lower bound of `C`. If it wasn't,
|
||||
-- then either `¬IsGLB A a` or `¬IsGLB B b`, a contradiction.
|
||||
by_contra nub
|
||||
have h : ¬IsGLB A a ∨ ¬IsGLB B b := by
|
||||
sorry
|
||||
cases h with
|
||||
| inl na => exact absurd ha na
|
||||
| inr nb => exact absurd hb nb
|
||||
|
||||
/--
|
||||
Theorem I.34
|
||||
|
||||
Given two nonempty subsets `S` and `T` of `ℝ` such that `s ≤ t` for every `s` in
|
||||
`S` and every `t` in `T`. Then `S` has a supremum, and `T` has an infimum, and
|
||||
they satisfy the inequality `sup S ≤ inf T`.
|
||||
-/
|
||||
theorem forall_mem_le_forall_mem_imp_sup_le_inf (S T : Set ℝ)
|
||||
(hS : S.Nonempty) (hT : T.Nonempty)
|
||||
(p : ∀ s ∈ S, ∀ t ∈ T, s ≤ t)
|
||||
: ∃ (s : ℝ), IsLUB S s ∧ ∃ (t : ℝ), IsGLB T t ∧ s ≤ t := by
|
||||
sorry
|
||||
|
||||
end Real
|
||||
|
|
|
|||
|
|
@ -74,10 +74,51 @@
|
|||
\ % Force space prior to *Proof.*
|
||||
|
||||
\begin{enumerate}[(a)]
|
||||
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.arb_close_to_sup}
|
||||
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.arb_close_to_inf}
|
||||
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.sup_imp_exists_gt_sup_sub_delta}
|
||||
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.inf_imp_exists_lt_inf_add_delta}
|
||||
\end{enumerate}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\begin{xtheorem}{I.33}[Additive Property]
|
||||
|
||||
Given nonempty subsets $A$ and $B$ of $\mathbb{R}$, let $C$ denote the set
|
||||
$$C = \{a + b : a \in A, b \in B\}.$$
|
||||
|
||||
\begin{enumerate}[(a)]
|
||||
\item If each of $A$ and $B$ has a supremum, then $C$ has a supremum, and
|
||||
$$\sup{C} = \sup{A} + \sup{B}.$$
|
||||
\item If each of $A$ and $B$ has an infimum, then $C$ has an infimum, and
|
||||
$$\inf{C} = \inf{A} + \inf{B}.$$
|
||||
\end{enumerate}
|
||||
|
||||
\end{xtheorem}
|
||||
|
||||
\begin{proof}
|
||||
|
||||
\ % Force space prior to *Proof.*
|
||||
|
||||
\begin{enumerate}[(a)]
|
||||
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.sup_minkowski_sum_eq_sup_add_sup}
|
||||
\item \href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.inf_minkowski_sum_eq_inf_add_inf}
|
||||
\end{enumerate}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\begin{xtheorem}{I.34}
|
||||
|
||||
Given two nonempty subsets $S$ and $T$ of $\mathbb{R}$ such that
|
||||
$$s \leq t$$
|
||||
for every $s$ in $S$ and every $t$ in $T$. Then $S$ has a supremum, and $T$
|
||||
has an infimum, and they satisfy the inequality
|
||||
$$\sup{S} \leq \inf{T}.$$
|
||||
|
||||
\end{xtheorem}
|
||||
|
||||
\begin{proof}
|
||||
|
||||
\href{Chapter_I_3.lean}{Apostol.Chapter_I_3.Real.forall_mem_le_forall_mem_imp_sup_le_inf}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\end{document}
|
||||
|
|
|
|||
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Reference in New Issue