English
The IccLeftChart provides a left chart between the closed interval Icc(x,y) and the one-dimensional Euclidean half-space, for x<y.
Русский
IccLeftChart задаёт левую карту между замкнутым интервалом Icc(x,y) и одномерным полупространством Евклида при x<y.
LaTeX
$$$\\text{IccLeftChart}(x,y) : \\text{OpenPartialHomeomorph}(\\mathrm{Icc}(x,y), \\mathrm{EuclideanHalfSpace}(1))$$$
Lean4
/-- The left chart for the topological space `[x, y]`, defined on `[x,y)` and sending `x` to `0` in
`EuclideanHalfSpace 1`.
-/
def IccLeftChart (x y : ℝ) [h : Fact (x < y)] : OpenPartialHomeomorph (Icc x y) (EuclideanHalfSpace 1)
where
source := {z : Icc x y | z.val < y}
target := {z : EuclideanHalfSpace 1 | z.val 0 < y - x}
toFun := fun z : Icc x y => ⟨fun _ => z.val - x, sub_nonneg.mpr z.property.1⟩
invFun z := ⟨min (z.val 0 + x) y, by simp [z.prop, h.out.le]⟩
map_source' := by simp only [imp_self, sub_lt_sub_iff_right, mem_setOf_eq, forall_true_iff]
map_target' := by
simp only [min_lt_iff, mem_setOf_eq]; intro z hz; left
linarith
left_inv' := by
rintro ⟨z, hz⟩ h'z
simp only [mem_setOf_eq, mem_Icc] at hz h'z
simp only [hz, min_eq_left, sub_add_cancel]
right_inv' := by
rintro ⟨z, hz⟩ h'z
rw [Subtype.mk_eq_mk]
funext i
dsimp at hz h'z
have A : x + z 0 ≤ y := by linarith
rw [Subsingleton.elim i 0]
simp only [A, add_comm, add_sub_cancel_left, min_eq_left]
open_source :=
haveI : IsOpen {z : ℝ | z < y} := isOpen_Iio
this.preimage continuous_subtype_val
open_target := by
have : IsOpen {z : ℝ | z < y - x} := isOpen_Iio
have : IsOpen {z : EuclideanSpace ℝ (Fin 1) | z 0 < y - x} :=
this.preimage (@continuous_apply (Fin 1) (fun _ => ℝ) _ 0)
exact this.preimage continuous_subtype_val
continuousOn_toFun := by
apply Continuous.continuousOn
apply Continuous.subtype_mk
have : Continuous fun (z : ℝ) (_ : Fin 1) => z - x :=
Continuous.sub (continuous_pi fun _ => continuous_id) continuous_const
exact this.comp continuous_subtype_val
continuousOn_invFun := by
apply Continuous.continuousOn
apply Continuous.subtype_mk
have A : Continuous fun z : ℝ => min (z + x) y := (continuous_id.add continuous_const).min continuous_const
have B : Continuous fun z : EuclideanSpace ℝ (Fin 1) => z 0 := continuous_apply 0
exact (A.comp B).comp continuous_subtype_val
