/-- Given a model with corners `(E, H)`, we define the groupoid of analytic transformations of `H`
as the maps that are analytic and map interior to interior when read in `E` through `I`. We also
explicitly define that they are `C^∞` on the whole domain, since we are only requiring
analyticity on the interior of the domain. -/
def analyticGroupoid : StructureGroupoid H :=
  (contDiffGroupoid ∞ I) ⊓ Pregroupoid.groupoid
    { property := fun f s => AnalyticOn 𝕜 (I ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ interior (range I)) ∧
        (I.symm ⁻¹' s ∩ interior (range I)).image (I ∘ f ∘ I.symm) ⊆ interior (range I)
      comp := fun {f g u v} hf hg _ _ _ => by
        simp only [] at hf hg ⊢
        have comp : I ∘ (g ∘ f) ∘ I.symm = (I ∘ g ∘ I.symm) ∘ I ∘ f ∘ I.symm := by ext x; simp
        apply And.intro
        · simp only [comp, preimage_inter]
          refine hg.left.comp (hf.left.mono ?_) ?_
          · simp only [subset_inter_iff, inter_subset_right]
            rw [inter_assoc]
            simp
          · intro x hx
            apply And.intro
            · rw [mem_preimage, comp_apply, I.left_inv]
              exact hx.left.right
            · apply hf.right
              rw [mem_image]
              exact ⟨x, ⟨⟨hx.left.left, hx.right⟩, rfl⟩⟩
        · simp only [comp]
          rw [image_comp]
          intro x hx
          rw [mem_image] at hx
          rcases hx with ⟨x', hx'⟩
          refine hg.right ⟨x', And.intro ?_ hx'.right⟩
          apply And.intro
          · have hx'1 : x' ∈ ((v.preimage f).preimage (I.symm)).image (I ∘ f ∘ I.symm) := by
              refine image_subset (I ∘ f ∘ I.symm) ?_ hx'.left
              rw [preimage_inter]
              refine Subset.trans ?_ (inter_subset_right (u.preimage I.symm)
                ((v.preimage f).preimage I.symm))
              apply inter_subset_left
            rcases hx'1 with ⟨x'', hx''⟩
            rw [hx''.right.symm]
            simp only [comp_apply, mem_preimage, I.left_inv]
            exact hx''.left
          · rw [mem_image] at hx'
            rcases hx'.left with ⟨x'', hx''⟩
            exact hf.right ⟨x'', ⟨⟨hx''.left.left.left, hx''.left.right⟩, hx''.right⟩⟩
      id_mem := by
        apply And.intro
        · simp only [preimage_univ, univ_inter]
          exact AnalyticOn.congr isOpen_interior
            (f := (1 : E →L[𝕜] E)) (fun x _ => (1 : E →L[𝕜] E).analyticAt x)
            (fun z hz => (I.right_inv (interior_subset hz)).symm)
        · intro x hx
          simp only [left_id, comp_apply, preimage_univ, univ_inter, mem_image] at hx
          rcases hx with ⟨y, hy⟩
          rw [← hy.right, I.right_inv (interior_subset hy.left)]
          exact hy.left
      locality := fun {f u} _ h => by
        simp only [] at h
        simp only [AnalyticOn]
        apply And.intro
        · intro x hx
          rcases h (I.symm x) (mem_preimage.mp hx.left) with ⟨v, hv⟩
          exact hv.right.right.left x ⟨mem_preimage.mpr ⟨hx.left, hv.right.left⟩, hx.right⟩
        · apply mapsTo'.mp
          simp only [MapsTo]
          intro x hx
          rcases h (I.symm x) hx.left with ⟨v, hv⟩
          apply hv.right.right.right
          rw [mem_image]
          have hx' := And.intro hx (mem_preimage.mpr hv.right.left)
          rw [← mem_inter_iff, inter_comm, ← inter_assoc, ← preimage_inter, inter_comm v u] at hx'
          exact ⟨x, ⟨hx', rfl⟩⟩
      congr := fun {f g u} hu fg hf => by
        simp only [] at hf ⊢
        apply And.intro
        · refine AnalyticOn.congr (IsOpen.inter (hu.preimage I.continuous_symm) isOpen_interior)
            hf.left ?_
          intro z hz
          simp only [comp_apply]
          rw [fg (I.symm z) hz.left]
        · intro x hx
          apply hf.right
          rw [mem_image] at hx ⊢
          rcases hx with ⟨y, hy⟩
          

Proof state:
case right.a.intro
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
f g : H → H
u : Set H
hu : IsOpen u
fg : ∀ x ∈ u, g x = f x
hf :
  AnalyticOn 𝕜 (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) (↑(ModelWithCorners.symm I) ⁻¹' u ∩ interior (range ↑I)) ∧
    ↑I ∘ f ∘ ↑(ModelWithCorners.symm I) '' (↑(ModelWithCorners.symm I) ⁻¹' u ∩ interior (range ↑I)) ⊆
      interior (range ↑I)
x y : E
hy : y ∈ ↑(ModelWithCorners.symm I) ⁻¹' u ∩ interior (range ↑I) ∧ (↑I ∘ g ∘ ↑(ModelWithCorners.symm I)) y = x
⊢ ∃ x_1 ∈ ↑(ModelWithCorners.symm I) ⁻¹' u ∩ interior (range ↑I), (↑I ∘ f ∘ ↑(ModelWithCorners.symm I)) x_1 = x

Next tactic:
refine ⟨y, ⟨hy.left, ?_⟩⟩