/-- The groupoid of `0`-times continuously differentiable maps is just the groupoid of all
local homeomorphisms -/
theorem contDiffGroupoid_zero_eq : contDiffGroupoid 0 I = continuousGroupoid H := by
  apply le_antisymm le_top
  intro u _
  -- we have to check that every local homeomorphism belongs to `contDiffGroupoid 0 I`,
  -- by unfolding its definition
  change u ∈ contDiffGroupoid 0 I
  rw [contDiffGroupoid, mem_groupoid_of_pregroupoid]
  

Proof state:
m n : ℕ∞
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
H : Type u_3
inst✝¹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝ : TopologicalSpace M
u : PartialHomeomorph H H
a✝ : u ∈ ⊤.members
⊢ Pregroupoid.property
      {
        property := fun f s =>
          ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),
        comp :=
          (_ :
            ∀ {f g : H → H} {u v : Set H},
              ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →
                ContDiffOn 𝕜 0 (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →
                  IsOpen u →
                    IsOpen v →
                      IsOpen (u ∩ f ⁻¹' v) →
                        (fun f s =>
                            ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                              (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))
                          (g ∘ f) (u ∩ f ⁻¹' v)),
        id_mem :=
          (_ : ContDiffOn 𝕜 0 (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),
        locality :=
          (_ :
            ∀ {f : H → H} {u : Set H},
              IsOpen u →
                (∀ x ∈ u,
                    ∃ v,
                      IsOpen v ∧
                        x ∈ v ∧
                          (fun f s =>
                              ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                                (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))
                            f (u ∩ v)) →
                  ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),
        congr :=
          (_ :
            ∀ {f g : H → H} {u : Set H},
              IsOpen u →
                (∀ x ∈ u, g x = f x) →
                  ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →
                    ContDiffOn 𝕜 0 (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))
                      (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }
      (↑u) u.source ∧
    Pregroupoid.property
      {
        property := fun f s =>
          ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I),
        comp :=
          (_ :
            ∀ {f g : H → H} {u v : Set H},
              ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →
                ContDiffOn 𝕜 0 (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' v ∩ range ↑I) →
                  IsOpen u →
                    IsOpen v →
                      IsOpen (u ∩ f ⁻¹' v) →
                        (fun f s =>
                            ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                              (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))
                          (g ∘ f) (u ∩ f ⁻¹' v)),
        id_mem :=
          (_ : ContDiffOn 𝕜 0 (↑I ∘ id ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' univ ∩ range ↑I)),
        locality :=
          (_ :
            ∀ {f : H → H} {u : Set H},
              IsOpen u →
                (∀ x ∈ u,
                    ∃ v,
                      IsOpen v ∧
                        x ∈ v ∧
                          (fun f s =>
                              ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I))
                                (↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I))
                            f (u ∩ v)) →
                  ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)),
        congr :=
          (_ :
            ∀ {f g : H → H} {u : Set H},
              IsOpen u →
                (∀ x ∈ u, g x = f x) →
                  ContDiffOn 𝕜 0 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I) →
                    ContDiffOn 𝕜 0 (↑I ∘ g ∘ ↑(ModelWithCorners.symm I))
                      (↑(ModelWithCorners.symm I) ⁻¹' u ∩ range ↑I)) }
      (↑(PartialHomeomorph.symm u)) u.target

Next tactic:
simp only [contDiffOn_zero]