27 lines
827 B
Plaintext
27 lines
827 B
Plaintext
|
import Mathlib.Data.Real.Sqrt
|
|||
|
import Mathlib.Logic.Function.Basic
|
|||
|
|
|||
|
namespace Real
|
|||
|
|
|||
|
notation "ℝ²" => ℝ × ℝ
|
|||
|
|
|||
|
noncomputable def dist (x y : ℝ²) :=
|
|||
|
Real.sqrt ((abs (y.1 - x.1)) ^ 2 + (abs (y.2 - x.2)) ^ 2)
|
|||
|
|
|||
|
def similar (S T : Set ℝ²) : Prop :=
|
|||
|
∃ f : ℝ² → ℝ², Function.Bijective f ∧
|
|||
|
∃ s : ℝ, ∀ x y : ℝ², x ∈ S ∧ y ∈ T →
|
|||
|
s * dist x y = dist (f x) (f y)
|
|||
|
|
|||
|
def congruent (S T : Set (ℝ × ℝ)) : Prop :=
|
|||
|
∃ f : ℝ² → ℝ², Function.Bijective f ∧
|
|||
|
∀ x y : ℝ², x ∈ S ∧ y ∈ T →
|
|||
|
dist x y = dist (f x) (f y)
|
|||
|
|
|||
|
theorem congruent_similar {S T : Set ℝ²} : congruent S T → similar S T := by
|
|||
|
intro hc
|
|||
|
let ⟨f, ⟨hf, hs⟩⟩ := hc
|
|||
|
conv at hs => intro x y hxy; arg 1; rw [← one_mul (dist x y)]
|
|||
|
exact ⟨f, ⟨hf, ⟨1, hs⟩⟩⟩
|
|||
|
|
|||
|
end Real
|