Restructure geometry and sequence modules further.
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import Mathlib.Data.Real.Sqrt
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import Mathlib.Logic.Function.Basic
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namespace Real
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notation "ℝ²" => ℝ × ℝ
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noncomputable def dist (x y : ℝ²) :=
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Real.sqrt ((abs (y.1 - x.1)) ^ 2 + (abs (y.2 - x.2)) ^ 2)
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def similar (S T : Set ℝ²) : Prop :=
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∃ f : ℝ² → ℝ², Function.Bijective f ∧
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∃ s : ℝ, ∀ x y : ℝ², x ∈ S ∧ y ∈ T →
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s * dist x y = dist (f x) (f y)
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def congruent (S T : Set (ℝ × ℝ)) : Prop :=
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∃ f : ℝ² → ℝ², Function.Bijective f ∧
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∀ x y : ℝ², x ∈ S ∧ y ∈ T →
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dist x y = dist (f x) (f y)
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theorem congruent_similar {S T : Set ℝ²} : congruent S T → similar S T := by
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intro hc
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let ⟨f, ⟨hf, hs⟩⟩ := hc
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conv at hs => intro x y hxy; arg 1; rw [← one_mul (dist x y)]
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exact ⟨f, ⟨hf, ⟨1, hs⟩⟩⟩
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end Real
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import Common.Data.Real.Geometry.Basic
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import Common.Data.Real.Geometry.Rectangle
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import Mathlib.Data.Real.Sqrt
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notation "ℝ²" => ℝ × ℝ
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namespace Real
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/--
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Determine the distance between two points in `ℝ²`.
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-/
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noncomputable def dist (x y : ℝ²) :=
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Real.sqrt ((abs (y.1 - x.1)) ^ 2 + (abs (y.2 - x.2)) ^ 2)
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/--
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Two sets `S` and `T` are `similar` iff there exists a one-to-one correspondence
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between `S` and `T` such that the distance between any two points `P, Q ∈ S` and
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corresponding points `P', Q' ∈ T` differ by some constant `α`. In other words,
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`α|PQ| = |P'Q'|`.
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-/
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def similar (S T : Set ℝ²) : Prop :=
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∃ f : ℝ² → ℝ², Function.Bijective f ∧
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∃ s : ℝ, ∀ x y : ℝ², x ∈ S ∧ y ∈ T →
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s * dist x y = dist (f x) (f y)
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/--
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Two sets are congruent if they are similar with a scaling factor of `1`.
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-/
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def congruent (S T : Set (ℝ × ℝ)) : Prop :=
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∃ f : ℝ² → ℝ², Function.Bijective f ∧
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∀ x y : ℝ², x ∈ S ∧ y ∈ T →
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dist x y = dist (f x) (f y)
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/--
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Any two congruent sets must be similar to one another.
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-/
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theorem congruent_similar {S T : Set ℝ²} : congruent S T → similar S T := by
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intro hc
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let ⟨f, ⟨hf, hs⟩⟩ := hc
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conv at hs => intro x y hxy; arg 1; rw [← one_mul (dist x y)]
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exact ⟨f, ⟨hf, ⟨1, hs⟩⟩⟩
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end Real
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import Common.Data.Real.Geometry.Basic
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namespace Real
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/--
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A `Rectangle` is characterized by two points defining opposite corners. We
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arbitrarily choose the bottom left and top right points to perform this
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characterization.
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-/
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structure Rectangle (bottom_left : ℝ²) (top_right : ℝ²)
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namespace Rectangle
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/--
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A `Rectangle` is the locus of points making up its edges.
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-/
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def set_eq (r : Rectangle x y) : Set ℝ² := sorry
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/--
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Computes the bottom right corner of a `Rectangle`.
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-/
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def bottom_right (r : Rectangle x y) : ℝ² := sorry
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/--
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Computes the top left corner of a `Rectangle`.
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-/
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def top_left (r : Rectangle x y) : ℝ² := sorry
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/--
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Computes the width of a `Rectangle`.
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-/
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def width (r : Rectangle x y) : ℝ := sorry
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/--
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Computes the height of a `Rectangle`.
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-/
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def height (r : Rectangle x y) : ℝ := sorry
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end Real.Rectangle
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@ -1,206 +1,2 @@
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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 `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 =>
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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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A term is equal to the next in the sequence minus the common difference.
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-/
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theorem term_closed_sub_succ_delta {seq : Arithmetic}
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: seq.termClosed n = seq.termClosed (n + 1) - seq.Δ :=
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calc
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seq.termClosed n
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_ = seq.a₀ + seq.Δ * n := rfl
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_ = seq.a₀ + seq.Δ * n + seq.Δ - seq.Δ := by rw [add_sub_cancel]
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_ = seq.a₀ + seq.Δ * (↑n + 1) - seq.Δ := by ring_nf
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_ = seq.a₀ + seq.Δ * ↑(n + 1) - seq.Δ := by simp only [Nat.cast_add, Nat.cast_one]
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_ = seq.termClosed (n + 1) - seq.Δ := 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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| 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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Simplify a summation of terms found in the proof of `sum_recursive_closed`.
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-/
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private lemma sub_delta_summand_eq_two_mul_a₀ {seq : Arithmetic}
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: seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ = 2 * seq.a₀ :=
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calc
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seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ
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_ = seq.a₀ + (seq.a₀ + seq.Δ * ↑(n + 1)) - (n + 1) * seq.Δ := rfl
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_ = seq.a₀ + seq.a₀ + seq.Δ * ↑(n + 1) - (n + 1) * seq.Δ := by rw [←add_assoc]
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_ = seq.a₀ + seq.a₀ + seq.Δ * (n + 1) - (n + 1) * seq.Δ := by simp only [Nat.cast_add, Nat.cast_one]
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_ = 2 * seq.a₀ := by ring_nf
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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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: seq.sum_recursive n = seq.sum_closed n := by
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induction n with
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| zero =>
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unfold sum_recursive sum_closed termClosed
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norm_num
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| succ n ih =>
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calc
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seq.sum_recursive (n + 1)
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_ = seq.termClosed (n + 1) + seq.sum_recursive n := rfl
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_ = seq.termClosed (n + 1) + seq.sum_closed n := by rw [ih]
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_ = seq.termClosed (n + 1) + ((n + 1) * (seq.a₀ + seq.termClosed n)) / 2 := rfl
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_ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed n + seq.a₀ + seq.termClosed n) / 2 := by ring_nf
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_ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * (seq.termClosed (n + 1) - seq.Δ) + seq.a₀ + (seq.termClosed (n + 1) - seq.Δ)) / 2 := by rw [@term_closed_sub_succ_delta n]
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_ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed (n + 1) + (seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ)) / 2 := by ring_nf
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_ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed (n + 1) + 2 * seq.a₀) / 2 := by rw [sub_delta_summand_eq_two_mul_a₀]
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_ = ((n + 1) + 1) * (seq.a₀ + seq.termClosed (n + 1)) / 2 := by ring_nf
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_ = (↑(n + 1) + 1) * (seq.a₀ + seq.termClosed (n + 1)) / 2 := by simp only [Nat.cast_add, Nat.cast_one]
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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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import Common.Data.Real.Sequence.Arithmetic
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import Common.Data.Real.Sequence.Geometric
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@ -0,0 +1,117 @@
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import Mathlib.Data.Real.Basic
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namespace Real
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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 =>
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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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A term is equal to the next in the sequence minus the common difference.
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-/
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theorem term_closed_sub_succ_delta {seq : Arithmetic}
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: seq.termClosed n = seq.termClosed (n + 1) - seq.Δ :=
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calc
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seq.termClosed n
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_ = seq.a₀ + seq.Δ * n := rfl
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_ = seq.a₀ + seq.Δ * n + seq.Δ - seq.Δ := by rw [add_sub_cancel]
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_ = seq.a₀ + seq.Δ * (↑n + 1) - seq.Δ := by ring_nf
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_ = seq.a₀ + seq.Δ * ↑(n + 1) - seq.Δ := by simp only [Nat.cast_add, Nat.cast_one]
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_ = seq.termClosed (n + 1) - seq.Δ := 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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| 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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Simplify a summation of terms found in the proof of `sum_recursive_closed`.
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-/
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private lemma sub_delta_summand_eq_two_mul_a₀ {seq : Arithmetic}
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: seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ = 2 * seq.a₀ :=
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calc
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seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ
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_ = seq.a₀ + (seq.a₀ + seq.Δ * ↑(n + 1)) - (n + 1) * seq.Δ := rfl
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_ = seq.a₀ + seq.a₀ + seq.Δ * ↑(n + 1) - (n + 1) * seq.Δ := by rw [←add_assoc]
|
||||
_ = seq.a₀ + seq.a₀ + seq.Δ * (n + 1) - (n + 1) * seq.Δ := by simp only [Nat.cast_add, Nat.cast_one]
|
||||
_ = 2 * seq.a₀ := by ring_nf
|
||||
|
||||
/--
|
||||
The recursive and closed definitions of the sum of an arithmetic sequence agree
|
||||
with one another.
|
||||
-/
|
||||
theorem sum_recursive_closed (seq : Arithmetic) (n : Nat)
|
||||
: seq.sum_recursive n = seq.sum_closed n := by
|
||||
induction n with
|
||||
| zero =>
|
||||
unfold sum_recursive sum_closed termClosed
|
||||
norm_num
|
||||
| succ n ih =>
|
||||
calc
|
||||
seq.sum_recursive (n + 1)
|
||||
_ = seq.termClosed (n + 1) + seq.sum_recursive n := rfl
|
||||
_ = seq.termClosed (n + 1) + seq.sum_closed n := by rw [ih]
|
||||
_ = seq.termClosed (n + 1) + ((n + 1) * (seq.a₀ + seq.termClosed n)) / 2 := rfl
|
||||
_ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed n + seq.a₀ + seq.termClosed n) / 2 := by ring_nf
|
||||
_ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * (seq.termClosed (n + 1) - seq.Δ) + seq.a₀ + (seq.termClosed (n + 1) - seq.Δ)) / 2 := by rw [@term_closed_sub_succ_delta n]
|
||||
_ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed (n + 1) + (seq.a₀ + seq.termClosed (n + 1) - (n + 1) * seq.Δ)) / 2 := by ring_nf
|
||||
_ = (2 * seq.termClosed (n + 1) + n * seq.a₀ + n * seq.termClosed (n + 1) + 2 * seq.a₀) / 2 := by rw [sub_delta_summand_eq_two_mul_a₀]
|
||||
_ = ((n + 1) + 1) * (seq.a₀ + seq.termClosed (n + 1)) / 2 := by ring_nf
|
||||
_ = (↑(n + 1) + 1) * (seq.a₀ + seq.termClosed (n + 1)) / 2 := by simp only [Nat.cast_add, Nat.cast_one]
|
||||
_ = seq.sum_closed (n + 1) := rfl
|
||||
|
||||
end Real.Arithmetic
|
||||
|
|
@ -0,0 +1,92 @@
|
|||
import Mathlib.Data.Real.Basic
|
||||
|
||||
namespace Real
|
||||
|
||||
/--
|
||||
A `0th`-indexed geometric sequence.
|
||||
-/
|
||||
structure Geometric where
|
||||
a₀ : Real
|
||||
r : Real
|
||||
|
||||
namespace Geometric
|
||||
|
||||
/--
|
||||
Returns the value of the `n`th term of a geometric sequence.
|
||||
|
||||
This function calculates the value of this term directly. Keep in mind the
|
||||
sequence is `0`th-indexed.
|
||||
-/
|
||||
def termClosed (seq : Geometric) (n : Nat) : Real :=
|
||||
seq.a₀ * seq.r ^ n
|
||||
|
||||
/--
|
||||
Returns the value of the `n`th term of a geometric sequence.
|
||||
|
||||
This function calculates the value of this term recursively. Keep in mind the
|
||||
sequence is `0`th-indexed.
|
||||
-/
|
||||
def termRecursive : Geometric → Nat → Real
|
||||
| seq, 0 => seq.a₀
|
||||
| seq, (n + 1) => seq.r * (seq.termRecursive n)
|
||||
|
||||
/--
|
||||
The recursive and closed term definitions of a geometric sequence agree with
|
||||
one another.
|
||||
-/
|
||||
theorem term_recursive_closed (seq : Geometric) (n : Nat)
|
||||
: seq.termRecursive n = seq.termClosed n := by
|
||||
induction n with
|
||||
| zero => unfold termClosed termRecursive; norm_num
|
||||
| succ n ih => calc
|
||||
seq.termRecursive (n + 1)
|
||||
_ = seq.r * (seq.termRecursive n) := rfl
|
||||
_ = seq.r * (seq.termClosed n) := by rw [ih]
|
||||
_ = seq.r * (seq.a₀ * seq.r ^ n) := rfl
|
||||
_ = seq.a₀ * seq.r ^ (n + 1) := by ring
|
||||
_ = seq.termClosed (n + 1) := rfl
|
||||
|
||||
/--
|
||||
The summation of the first `n + 1` terms of a geometric sequence.
|
||||
|
||||
This function calculates the sum directly.
|
||||
-/
|
||||
noncomputable def sum_closed_ratio_neq_one (seq : Geometric) (n : Nat)
|
||||
: seq.r ≠ 1 → Real :=
|
||||
fun _ => (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r)
|
||||
|
||||
/--
|
||||
The summation of the first `n + 1` terms of a geometric sequence.
|
||||
|
||||
This function calculates the sum recursively.
|
||||
-/
|
||||
def sum_recursive : Geometric → Nat → Real
|
||||
| seq, 0 => seq.a₀
|
||||
| seq, (n + 1) => seq.termClosed (n + 1) + seq.sum_recursive n
|
||||
|
||||
/--
|
||||
The recursive and closed definitions of the sum of a geometric sequence agree
|
||||
with one another.
|
||||
-/
|
||||
theorem sum_recursive_closed (seq : Geometric) (n : Nat) (p : seq.r ≠ 1)
|
||||
: sum_recursive seq n = sum_closed_ratio_neq_one seq n p := by
|
||||
have h : 1 - seq.r ≠ 0 := by
|
||||
intro h
|
||||
rw [sub_eq_iff_eq_add, zero_add] at h
|
||||
exact False.elim (p (Eq.symm h))
|
||||
induction n with
|
||||
| zero =>
|
||||
unfold sum_recursive sum_closed_ratio_neq_one
|
||||
simp
|
||||
rw [mul_div_assoc, div_self h, mul_one]
|
||||
| succ n ih =>
|
||||
calc
|
||||
sum_recursive seq (n + 1)
|
||||
_ = seq.termClosed (n + 1) + seq.sum_recursive n := rfl
|
||||
_ = seq.termClosed (n + 1) + sum_closed_ratio_neq_one seq n p := by rw [ih]
|
||||
_ = seq.a₀ * seq.r ^ (n + 1) + (seq.a₀ * (1 - seq.r ^ (n + 1))) / (1 - seq.r) := rfl
|
||||
_ = 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]
|
||||
_ = (seq.a₀ * (1 - seq.r ^ (n + 1 + 1))) / (1 - seq.r) := by ring_nf
|
||||
_ = sum_closed_ratio_neq_one seq (n + 1) p := rfl
|
||||
|
||||
end Real.Geometric
|
||||
Loading…
Reference in New Issue