/
real.lean
43 lines (32 loc) · 1.06 KB
/
real.lean
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import challenge_notations
import challenge_prerequisites
import algebra.order.complete_field
/-!
This file shows that `ℝ` is a conditionally complete linearly ordered field.
Key result from mathlib:
[#3292](https://github.com/leanprover-community/mathlib/pull/3292)
-/
noncomputable theory
open_locale liquid_tensor_experiment nnreal zero_object
open liquid_tensor_experiment category_theory category_theory.limits opposite
/-!
The real numbers `ℝ` are a conditionally complete linearly ordered field.
-/
example : conditionally_complete_linear_ordered_field ℝ :=
infer_instance
/-!
The type of nonnegative real numbers is denoted by `ℝ≥0`.
-/
example : ℝ≥0 = {r : ℝ // r ≥ 0} :=
rfl
/-!
Any conditionally complete linear ordered field is isomorphic (as an ordered ring) to `ℝ`.
-/
example {R : Type} [conditionally_complete_linear_ordered_field R] : R ≃+*o ℝ :=
default
/-!
The isomorphism above is unique.
-/
example {R : Type} [conditionally_complete_linear_ordered_field R] (e₁ e₂ : R ≃+*o ℝ) :
e₁ = e₂ :=
subsingleton.elim _ _