41 lines
1.2 KiB
Plaintext
41 lines
1.2 KiB
Plaintext
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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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