[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Geometry.Manifold.ContMDiff.Defs).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem contMDiffWithinAt_iff_of_mem_source' {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source)
    (hy : f x' ∈ (chartAt H' y).source) :
    ContMDiffWithinAt I I' n f s x' ↔
      ContinuousWithinAt f s x' ∧
        ContDiffWithinAt 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
          ((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
          (extChartAt I x x')
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContDiffWithinAtProp (L79-L81)
def ContDiffWithinAtProp (n : WithTop ℕ∞) (f : H → H') (s : Set H) (x : H) : Prop :=
ContDiffWithinAt 𝕜 n (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x)
```

-- snippet 2
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_self_source (L82-L86)
theorem contDiffWithinAtProp_self_source {f : E → H'} {s : Set E} {x : E} :
    ContDiffWithinAtProp 𝓘(𝕜, E) I' n f s x ↔ ContDiffWithinAt 𝕜 n (I' ∘ f) s x := by
simp_rw [ContDiffWithinAtProp, modelWithCornersSelf_coe, range_id, inter_univ,
    modelWithCornersSelf_coe_symm, CompTriple.comp_eq, preimage_id_eq, id_eq]
```

-- snippet 3
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_self (L87-L90)
theorem contDiffWithinAtProp_self {f : E → E'} {s : Set E} {x : E} :
    ContDiffWithinAtProp 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ ContDiffWithinAt 𝕜 n f s x :=
contDiffWithinAtProp_self_source
```

-- snippet 4
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_self_target (L91-L97)
theorem contDiffWithinAtProp_self_target {f : H → E'} {s : Set H} {x : H} :
    ContDiffWithinAtProp I 𝓘(𝕜, E') n f s x ↔
      ContDiffWithinAt 𝕜 n (f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x) :=
Iff.rfl

/-- Being `Cⁿ` in the model space is a local property, invariant under `Cⁿ` maps. Therefore,
it lifts nicely to manifolds. -/
```

-- snippet 5
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAt_localInvariantProp_of_le (L98-L140)
theorem contDiffWithinAt_localInvariantProp_of_le (n m : WithTop ℕ∞) (hmn : m ≤ n) :
    (contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
      (ContDiffWithinAtProp I I' m) where
  is_local {s x u f} u_open xu := by
have : I.symm ⁻¹' (s ∩ u) ∩ range I = I.symm ⁻¹' s ∩ range I ∩ I.symm ⁻¹' u := by
      simp only [inter_right_comm, preimage_inter]
    rw [ContDiffWithinAtProp, ContDiffWithinAtProp, this]
    symm
    apply contDiffWithinAt_inter
    have : u ∈ 𝓝 (I.symm (I x)) := by
      rw [ModelWithCorners.left_inv]
      exact u_open.mem_nhds xu
    apply ContinuousAt.preimage_mem_nhds I.continuous_symm.continuousAt this
  right_invariance' {s x f e} he hx h := by
    rw [ContDiffWithinAtProp] at h ⊢
    have : I x = (I ∘ e.symm ∘ I.symm) (I (e x)) := by simp only [hx, mfld_simps]
    rw [this] at h
    have : I (e x) ∈ I.symm ⁻¹' e.target ∩ range I := by simp only [hx, mfld_simps]
    have := (mem_groupoid_of_pregroupoid.2 he).2.contDiffWithinAt this
    convert (h.comp_inter _ (this.of_le hmn)).mono_of_mem_nhdsWithin _
      using 1
    · ext y; simp only [mfld_simps]
    refine mem_nhdsWithin.mpr
      ⟨I.symm ⁻¹' e.target, e.open_target.preimage I.continuous_symm, by
        simp_rw [mem_preimage, I.left_inv, e.mapsTo hx], ?_⟩
    mfld_set_tac
  congr_of_forall {s x f g} h hx hf := by
    apply hf.congr
    · intro y hy
      simp only [mfld_simps] at hy
      simp only [h, hy, mfld_simps]
    · simp only [hx, mfld_simps]
  left_invariance' {s x f e'} he' hs hx h := by
    rw [ContDiffWithinAtProp] at h ⊢
    have A : (I' ∘ f ∘ I.symm) (I x) ∈ I'.symm ⁻¹' e'.source ∩ range I' := by
      simp only [hx, mfld_simps]
    have := (mem_groupoid_of_pregroupoid.2 he').1.contDiffWithinAt A
    convert (this.of_le hmn).comp _ h _
    · ext y; simp only [mfld_simps]
    · intro y hy; simp only [mfld_simps] at hy; simpa only [hy, mfld_simps] using hs hy.1

/-- Being `Cⁿ` in the model space is a local property, invariant under `C^n` maps. Therefore,
it lifts nicely to manifolds. -/
```

-- snippet 6
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAt_localInvariantProp (L141-L145)
theorem contDiffWithinAt_localInvariantProp (n : WithTop ℕ∞) :
    (contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
      (ContDiffWithinAtProp I I' n) :=
contDiffWithinAt_localInvariantProp_of_le n n le_rfl
```

-- snippet 7
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_mono_of_mem_nhdsWithin (L146-L152)
theorem contDiffWithinAtProp_mono_of_mem_nhdsWithin
    (n : WithTop ℕ∞) ⦃s x t⦄ ⦃f : H → H'⦄ (hts : s ∈ 𝓝[t] x)
    (h : ContDiffWithinAtProp I I' n f s x) : ContDiffWithinAtProp I I' n f t x := by
refine h.mono_of_mem_nhdsWithin ?_
  refine inter_mem ?_ (mem_of_superset self_mem_nhdsWithin inter_subset_right)
  rwa [← Filter.mem_map, ← I.image_eq, I.symm_map_nhdsWithin_image]
```

-- snippet 8
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_id (L153-L163)
theorem contDiffWithinAtProp_id (x : H) : ContDiffWithinAtProp I I n id univ x := by
simp only [ContDiffWithinAtProp, id_comp, preimage_univ, univ_inter]
  have : ContDiffWithinAt 𝕜 n id (range I) (I x) := contDiff_id.contDiffAt.contDiffWithinAt
  refine this.congr (fun y hy => ?_) ?_
  · simp only [ModelWithCorners.right_inv I hy, mfld_simps]
  · simp only [mfld_simps]

variable (I I') in
/-- A function is `n` times continuously differentiable within a set at a point in a manifold if
it is continuous and it is `n` times continuously differentiable in this set around this point, when
read in the preferred chart at this point. -/
```

-- snippet 9
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffWithinAt (L164-L170)
def ContMDiffWithinAt (n : WithTop ℕ∞) (f : M → M') (s : Set M) (x : M) :=
LiftPropWithinAt (ContDiffWithinAtProp I I' n) f s x

variable (I I') in
/-- A function is `n` times continuously differentiable at a point in a manifold if
it is continuous and it is `n` times continuously differentiable around this point, when
read in the preferred chart at this point. -/
```

-- snippet 10
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffAt (L171-L173)
def ContMDiffAt (n : WithTop ℕ∞) (f : M → M') (x : M) :=
ContMDiffWithinAt I I' n f univ x
```

-- snippet 11
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contMDiffAt_iff (L174-L184)
theorem contMDiffAt_iff {n : WithTop ℕ∞} {f : M → M'} {x : M} :
    ContMDiffAt I I' n f x ↔
      ContinuousAt f x ∧
        ContDiffWithinAt 𝕜 n (extChartAt I' (f x) ∘ f ∘ (extChartAt I x).symm) (range I)
          (extChartAt I x x) :=
liftPropAt_iff.trans <| by rw [ContDiffWithinAtProp, preimage_univ, univ_inter]; rfl

variable (I I') in
/-- A function is `n` times continuously differentiable in a set of a manifold if it is continuous
and, for any pair of points, it is `n` times continuously differentiable on this set in the charts
around these points. -/
```

-- snippet 12
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffOn (L185-L191)
def ContMDiffOn (n : WithTop ℕ∞) (f : M → M') (s : Set M) :=
∀ x ∈ s, ContMDiffWithinAt I I' n f s x

variable (I I') in
/-- A function is `n` times continuously differentiable in a manifold if it is continuous
and, for any pair of points, it is `n` times continuously differentiable in the charts
around these points. -/
```

-- snippet 13
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiff (L192-L196)
def ContMDiff (n : WithTop ℕ∞) (f : M → M') :=
∀ x, ContMDiffAt I I' n f x

/-! ### Deducing smoothness from higher smoothness -/
```

-- snippet 14
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffWithinAt (L197-L201)
theorem ContMDiffWithinAt.of_le (hf : ContMDiffWithinAt I I' n f s x) (le : m ≤ n) :
    ContMDiffWithinAt I I' m f s x := by
simp only [ContMDiffWithinAt] at hf ⊢
  exact ⟨hf.1, hf.2.of_le (mod_cast le)⟩
```

-- snippet 15
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffAt (L202-L204)
theorem ContMDiffAt.of_le (hf : ContMDiffAt I I' n f x) (le : m ≤ n) : ContMDiffAt I I' m f x :=
ContMDiffWithinAt.of_le hf le
```

-- snippet 16
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.leibniz (L107-L109)
theorem leibniz : X (f * f') = f • X f' + f' • X f :=
X.leibniz' _ _
```

-- snippet 17
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt (L189-L193)
def evalAt : LeftInvariantDerivation I G →ₗ[𝕜] PointDerivation I g where
  toFun X := Derivation.evalAt g X.1
map_add' _ _ := rfl
  map_smul' _ _ := rfl
```

-- snippet 18
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_neg (L136-L138)
theorem coe_neg : ⇑(-X) = -X :=
rfl
```

-- snippet 19
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_mul (L205-L211)
theorem evalAt_mul : evalAt (g * h) X = 𝒅ₕ (L_apply I g h) (evalAt h X) := by
ext f
  rw [← left_invariant, hfdifferential_apply, hfdifferential_apply, L_mul, fdifferential_comp,
    fdifferential_apply]
  simp only [ContMDiffMap.comp_apply, LinearMap.comp_apply]
  rw [fdifferential_apply, ← hfdifferential_apply (smoothLeftMul_one I h), left_invariant]
```

-- snippet 20
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_sub (L140-L142)
theorem coe_sub : ⇑(X - Y) = X - Y :=
rfl
```

-- snippet 21
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coeFnAddMonoidHom (L179-L183)
def coeFnAddMonoidHom : LeftInvariantDerivation I G →+ C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯ :=
⟨⟨DFunLike.coe, coe_zero⟩, coe_add⟩

variable {I G}
```

-- snippet 22
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.commutator_coe_derivation (L231-L236)
theorem commutator_coe_derivation :
    ⇑⁅X, Y⁆ =
      (⁅(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯), Y⁆ :
        Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) :=
rfl
```

-- snippet 23
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_smul (L172-L177)
theorem lift_smul (k : 𝕜) : (k • X).1 = k • X.1 :=
rfl

variable (I G)

/-- The coercion to function is a monoid homomorphism. -/
```

-- snippet 24
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.left_invariant (L201-L204)
theorem left_invariant : 𝒅ₕ (smoothLeftMul_one I g) (evalAt (1 : G) X) = evalAt g X :=
X.left_invariant'' g

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 25
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.hasNatScalar (L152-L154)
instance hasNatScalar : SMul ℕ (LeftInvariantDerivation I G) where
  smul r X := ⟨r • X.1, fun g => by simp_rw [LinearMap.map_smul_of_tower _ r, left_invariant']⟩
```

-- snippet 26
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.ext (L83-L86)
theorem ext (h : ∀ f, X f = Y f) : X = Y := DFunLike.ext _ _ h
variable (X Y f)
```

-- snippet 27
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_derivation (L87-L91)
theorem coe_derivation :
    ⇑(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = (X : C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯) :=
rfl

/-- Premature version of the lemma. Prefer using `left_invariant` instead. -/
```

-- snippet 28
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_zero (L132-L134)
theorem coe_zero : ⇑(0 : LeftInvariantDerivation I G) = 0 :=
rfl
```

-- snippet 29
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_coe (L198-L200)
theorem evalAt_coe : Derivation.evalAt g ↑X = evalAt g X :=
rfl
```

-- snippet 30
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_add (L128-L130)
theorem coe_add : ⇑(X + Y) = X + Y :=
rfl
```

-- snippet 31
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_smul (L168-L170)
theorem coe_smul : ⇑(r • X) = r • ⇑X :=
rfl
```

-- snippet 32
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.left_invariant' (L92-L105)
theorem left_invariant' :
    𝒅ₕ (smoothLeftMul_one I g) (Derivation.evalAt (1 : G) ↑X) = Derivation.evalAt g ↑X :=
left_invariant'' X g

protected theorem map_add : X (f + f') = X f + X f' := by simp

protected theorem map_zero : X 0 = 0 := by simp

protected theorem map_neg : X (-f) = -X f := by simp

protected theorem map_sub : X (f - f') = X f - X f' := by simp

protected theorem map_smul : X (r • f) = r • X f := by simp
```

-- snippet 33
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.toFun_eq_coe (L75-L77)
theorem toFun_eq_coe : X.toFun = ⇑X :=
rfl
```

-- snippet 34
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.hasIntScalar (L155-L157)
instance hasIntScalar : SMul ℤ (LeftInvariantDerivation I G) where
  smul r X := ⟨r • X.1, fun g => by simp_rw [LinearMap.map_smul_of_tower _ r, left_invariant']⟩
```

-- snippet 35
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_zero (L148-L151)
theorem lift_zero :
    (↑(0 : LeftInvariantDerivation I G) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = 0 :=
rfl
```

-- snippet 36
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.toDerivation_injective (L61-L64)
theorem toDerivation_injective :
    Function.Injective (toDerivation : LeftInvariantDerivation I G → _) :=
fun X Y h => by cases X; cases Y; congr
```

-- snippet 37
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_apply (L194-L196)
theorem evalAt_apply : evalAt g X f = (X f) g :=
rfl
```

-- snippet 38
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_add (L144-L146)
theorem lift_add : (↑(X + Y) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = X + Y :=
rfl
```

-- snippet 39
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.comp_L (L212-L216)
theorem comp_L : (X f).comp (𝑳 I g) = X (f.comp (𝑳 I g)) := by
ext h
  rw [ContMDiffMap.comp_apply, L_apply, ← evalAt_apply, evalAt_mul, hfdifferential_apply,
    fdifferential_apply, evalAt_apply]
```

-- snippet 40
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_injective (L78-L81)
theorem coe_injective :
    @Function.Injective (LeftInvariantDerivation I G) (_ → C^∞⟮I, G; 𝕜⟯) DFunLike.coe :=
DFunLike.coe_injective
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Geometry.Manifold.ContMDiff.Defs).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem contMDiffWithinAt_iff_of_mem_source' {x' : M} {y : M'} (hx : x' ∈ (chartAt H x).source)
    (hy : f x' ∈ (chartAt H' y).source) :
    ContMDiffWithinAt I I' n f s x' ↔
      ContinuousWithinAt f s x' ∧
        ContDiffWithinAt 𝕜 n (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
          ((extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
          (extChartAt I x x')
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContDiffWithinAtProp (L79-L81)
def ContDiffWithinAtProp (n : WithTop ℕ∞) (f : H → H') (s : Set H) (x : H) : Prop :=
ContDiffWithinAt 𝕜 n (I' ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x)
```

-- snippet 2
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_self_source (L82-L86)
theorem contDiffWithinAtProp_self_source {f : E → H'} {s : Set E} {x : E} :
    ContDiffWithinAtProp 𝓘(𝕜, E) I' n f s x ↔ ContDiffWithinAt 𝕜 n (I' ∘ f) s x := by
simp_rw [ContDiffWithinAtProp, modelWithCornersSelf_coe, range_id, inter_univ,
    modelWithCornersSelf_coe_symm, CompTriple.comp_eq, preimage_id_eq, id_eq]
```

-- snippet 3
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_self (L87-L90)
theorem contDiffWithinAtProp_self {f : E → E'} {s : Set E} {x : E} :
    ContDiffWithinAtProp 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ ContDiffWithinAt 𝕜 n f s x :=
contDiffWithinAtProp_self_source
```

-- snippet 4
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_self_target (L91-L97)
theorem contDiffWithinAtProp_self_target {f : H → E'} {s : Set H} {x : H} :
    ContDiffWithinAtProp I 𝓘(𝕜, E') n f s x ↔
      ContDiffWithinAt 𝕜 n (f ∘ I.symm) (I.symm ⁻¹' s ∩ range I) (I x) :=
Iff.rfl

/-- Being `Cⁿ` in the model space is a local property, invariant under `Cⁿ` maps. Therefore,
it lifts nicely to manifolds. -/
```

-- snippet 5
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAt_localInvariantProp_of_le (L98-L140)
theorem contDiffWithinAt_localInvariantProp_of_le (n m : WithTop ℕ∞) (hmn : m ≤ n) :
    (contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
      (ContDiffWithinAtProp I I' m) where
  is_local {s x u f} u_open xu := by
have : I.symm ⁻¹' (s ∩ u) ∩ range I = I.symm ⁻¹' s ∩ range I ∩ I.symm ⁻¹' u := by
      simp only [inter_right_comm, preimage_inter]
    rw [ContDiffWithinAtProp, ContDiffWithinAtProp, this]
    symm
    apply contDiffWithinAt_inter
    have : u ∈ 𝓝 (I.symm (I x)) := by
      rw [ModelWithCorners.left_inv]
      exact u_open.mem_nhds xu
    apply ContinuousAt.preimage_mem_nhds I.continuous_symm.continuousAt this
  right_invariance' {s x f e} he hx h := by
    rw [ContDiffWithinAtProp] at h ⊢
    have : I x = (I ∘ e.symm ∘ I.symm) (I (e x)) := by simp only [hx, mfld_simps]
    rw [this] at h
    have : I (e x) ∈ I.symm ⁻¹' e.target ∩ range I := by simp only [hx, mfld_simps]
    have := (mem_groupoid_of_pregroupoid.2 he).2.contDiffWithinAt this
    convert (h.comp_inter _ (this.of_le hmn)).mono_of_mem_nhdsWithin _
      using 1
    · ext y; simp only [mfld_simps]
    refine mem_nhdsWithin.mpr
      ⟨I.symm ⁻¹' e.target, e.open_target.preimage I.continuous_symm, by
        simp_rw [mem_preimage, I.left_inv, e.mapsTo hx], ?_⟩
    mfld_set_tac
  congr_of_forall {s x f g} h hx hf := by
    apply hf.congr
    · intro y hy
      simp only [mfld_simps] at hy
      simp only [h, hy, mfld_simps]
    · simp only [hx, mfld_simps]
  left_invariance' {s x f e'} he' hs hx h := by
    rw [ContDiffWithinAtProp] at h ⊢
    have A : (I' ∘ f ∘ I.symm) (I x) ∈ I'.symm ⁻¹' e'.source ∩ range I' := by
      simp only [hx, mfld_simps]
    have := (mem_groupoid_of_pregroupoid.2 he').1.contDiffWithinAt A
    convert (this.of_le hmn).comp _ h _
    · ext y; simp only [mfld_simps]
    · intro y hy; simp only [mfld_simps] at hy; simpa only [hy, mfld_simps] using hs hy.1

/-- Being `Cⁿ` in the model space is a local property, invariant under `C^n` maps. Therefore,
it lifts nicely to manifolds. -/
```

-- snippet 6
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAt_localInvariantProp (L141-L145)
theorem contDiffWithinAt_localInvariantProp (n : WithTop ℕ∞) :
    (contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
      (ContDiffWithinAtProp I I' n) :=
contDiffWithinAt_localInvariantProp_of_le n n le_rfl
```

-- snippet 7
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_mono_of_mem_nhdsWithin (L146-L152)
theorem contDiffWithinAtProp_mono_of_mem_nhdsWithin
    (n : WithTop ℕ∞) ⦃s x t⦄ ⦃f : H → H'⦄ (hts : s ∈ 𝓝[t] x)
    (h : ContDiffWithinAtProp I I' n f s x) : ContDiffWithinAtProp I I' n f t x := by
refine h.mono_of_mem_nhdsWithin ?_
  refine inter_mem ?_ (mem_of_superset self_mem_nhdsWithin inter_subset_right)
  rwa [← Filter.mem_map, ← I.image_eq, I.symm_map_nhdsWithin_image]
```

-- snippet 8
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contDiffWithinAtProp_id (L153-L163)
theorem contDiffWithinAtProp_id (x : H) : ContDiffWithinAtProp I I n id univ x := by
simp only [ContDiffWithinAtProp, id_comp, preimage_univ, univ_inter]
  have : ContDiffWithinAt 𝕜 n id (range I) (I x) := contDiff_id.contDiffAt.contDiffWithinAt
  refine this.congr (fun y hy => ?_) ?_
  · simp only [ModelWithCorners.right_inv I hy, mfld_simps]
  · simp only [mfld_simps]

variable (I I') in
/-- A function is `n` times continuously differentiable within a set at a point in a manifold if
it is continuous and it is `n` times continuously differentiable in this set around this point, when
read in the preferred chart at this point. -/
```

-- snippet 9
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffWithinAt (L164-L170)
def ContMDiffWithinAt (n : WithTop ℕ∞) (f : M → M') (s : Set M) (x : M) :=
LiftPropWithinAt (ContDiffWithinAtProp I I' n) f s x

variable (I I') in
/-- A function is `n` times continuously differentiable at a point in a manifold if
it is continuous and it is `n` times continuously differentiable around this point, when
read in the preferred chart at this point. -/
```

-- snippet 10
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffAt (L171-L173)
def ContMDiffAt (n : WithTop ℕ∞) (f : M → M') (x : M) :=
ContMDiffWithinAt I I' n f univ x
```

-- snippet 11
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.contMDiffAt_iff (L174-L184)
theorem contMDiffAt_iff {n : WithTop ℕ∞} {f : M → M'} {x : M} :
    ContMDiffAt I I' n f x ↔
      ContinuousAt f x ∧
        ContDiffWithinAt 𝕜 n (extChartAt I' (f x) ∘ f ∘ (extChartAt I x).symm) (range I)
          (extChartAt I x x) :=
liftPropAt_iff.trans <| by rw [ContDiffWithinAtProp, preimage_univ, univ_inter]; rfl

variable (I I') in
/-- A function is `n` times continuously differentiable in a set of a manifold if it is continuous
and, for any pair of points, it is `n` times continuously differentiable on this set in the charts
around these points. -/
```

-- snippet 12
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffOn (L185-L191)
def ContMDiffOn (n : WithTop ℕ∞) (f : M → M') (s : Set M) :=
∀ x ∈ s, ContMDiffWithinAt I I' n f s x

variable (I I') in
/-- A function is `n` times continuously differentiable in a manifold if it is continuous
and, for any pair of points, it is `n` times continuously differentiable in the charts
around these points. -/
```

-- snippet 13
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiff (L192-L196)
def ContMDiff (n : WithTop ℕ∞) (f : M → M') :=
∀ x, ContMDiffAt I I' n f x

/-! ### Deducing smoothness from higher smoothness -/
```

-- snippet 14
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffWithinAt (L197-L201)
theorem ContMDiffWithinAt.of_le (hf : ContMDiffWithinAt I I' n f s x) (le : m ≤ n) :
    ContMDiffWithinAt I I' m f s x := by
simp only [ContMDiffWithinAt] at hf ⊢
  exact ⟨hf.1, hf.2.of_le (mod_cast le)⟩
```

-- snippet 15
```lean
-- Mathlib.Geometry.Manifold.ContMDiff.Defs.ContMDiffAt (L202-L204)
theorem ContMDiffAt.of_le (hf : ContMDiffAt I I' n f x) (le : m ≤ n) : ContMDiffAt I I' m f x :=
ContMDiffWithinAt.of_le hf le
```

-- snippet 16
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.leibniz (L107-L109)
theorem leibniz : X (f * f') = f • X f' + f' • X f :=
X.leibniz' _ _
```

-- snippet 17
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt (L189-L193)
def evalAt : LeftInvariantDerivation I G →ₗ[𝕜] PointDerivation I g where
  toFun X := Derivation.evalAt g X.1
map_add' _ _ := rfl
  map_smul' _ _ := rfl
```

-- snippet 18
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_neg (L136-L138)
theorem coe_neg : ⇑(-X) = -X :=
rfl
```

-- snippet 19
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_mul (L205-L211)
theorem evalAt_mul : evalAt (g * h) X = 𝒅ₕ (L_apply I g h) (evalAt h X) := by
ext f
  rw [← left_invariant, hfdifferential_apply, hfdifferential_apply, L_mul, fdifferential_comp,
    fdifferential_apply]
  simp only [ContMDiffMap.comp_apply, LinearMap.comp_apply]
  rw [fdifferential_apply, ← hfdifferential_apply (smoothLeftMul_one I h), left_invariant]
```

-- snippet 20
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_sub (L140-L142)
theorem coe_sub : ⇑(X - Y) = X - Y :=
rfl
```

-- snippet 21
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coeFnAddMonoidHom (L179-L183)
def coeFnAddMonoidHom : LeftInvariantDerivation I G →+ C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯ :=
⟨⟨DFunLike.coe, coe_zero⟩, coe_add⟩

variable {I G}
```

-- snippet 22
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.commutator_coe_derivation (L231-L236)
theorem commutator_coe_derivation :
    ⇑⁅X, Y⁆ =
      (⁅(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯), Y⁆ :
        Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) :=
rfl
```

-- snippet 23
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_smul (L172-L177)
theorem lift_smul (k : 𝕜) : (k • X).1 = k • X.1 :=
rfl

variable (I G)

/-- The coercion to function is a monoid homomorphism. -/
```

-- snippet 24
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.left_invariant (L201-L204)
theorem left_invariant : 𝒅ₕ (smoothLeftMul_one I g) (evalAt (1 : G) X) = evalAt g X :=
X.left_invariant'' g

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 25
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.hasNatScalar (L152-L154)
instance hasNatScalar : SMul ℕ (LeftInvariantDerivation I G) where
  smul r X := ⟨r • X.1, fun g => by simp_rw [LinearMap.map_smul_of_tower _ r, left_invariant']⟩
```

-- snippet 26
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.ext (L83-L86)
theorem ext (h : ∀ f, X f = Y f) : X = Y := DFunLike.ext _ _ h
variable (X Y f)
```

-- snippet 27
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_derivation (L87-L91)
theorem coe_derivation :
    ⇑(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = (X : C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯) :=
rfl

/-- Premature version of the lemma. Prefer using `left_invariant` instead. -/
```

-- snippet 28
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_zero (L132-L134)
theorem coe_zero : ⇑(0 : LeftInvariantDerivation I G) = 0 :=
rfl
```

-- snippet 29
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_coe (L198-L200)
theorem evalAt_coe : Derivation.evalAt g ↑X = evalAt g X :=
rfl
```

-- snippet 30
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_add (L128-L130)
theorem coe_add : ⇑(X + Y) = X + Y :=
rfl
```

-- snippet 31
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_smul (L168-L170)
theorem coe_smul : ⇑(r • X) = r • ⇑X :=
rfl
```

-- snippet 32
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.left_invariant' (L92-L105)
theorem left_invariant' :
    𝒅ₕ (smoothLeftMul_one I g) (Derivation.evalAt (1 : G) ↑X) = Derivation.evalAt g ↑X :=
left_invariant'' X g

protected theorem map_add : X (f + f') = X f + X f' := by simp

protected theorem map_zero : X 0 = 0 := by simp

protected theorem map_neg : X (-f) = -X f := by simp

protected theorem map_sub : X (f - f') = X f - X f' := by simp

protected theorem map_smul : X (r • f) = r • X f := by simp
```

-- snippet 33
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.toFun_eq_coe (L75-L77)
theorem toFun_eq_coe : X.toFun = ⇑X :=
rfl
```

-- snippet 34
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.hasIntScalar (L155-L157)
instance hasIntScalar : SMul ℤ (LeftInvariantDerivation I G) where
  smul r X := ⟨r • X.1, fun g => by simp_rw [LinearMap.map_smul_of_tower _ r, left_invariant']⟩
```

-- snippet 35
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_zero (L148-L151)
theorem lift_zero :
    (↑(0 : LeftInvariantDerivation I G) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = 0 :=
rfl
```

-- snippet 36
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.toDerivation_injective (L61-L64)
theorem toDerivation_injective :
    Function.Injective (toDerivation : LeftInvariantDerivation I G → _) :=
fun X Y h => by cases X; cases Y; congr
```

-- snippet 37
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_apply (L194-L196)
theorem evalAt_apply : evalAt g X f = (X f) g :=
rfl
```

-- snippet 38
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_add (L144-L146)
theorem lift_add : (↑(X + Y) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = X + Y :=
rfl
```

-- snippet 39
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.comp_L (L212-L216)
theorem comp_L : (X f).comp (𝑳 I g) = X (f.comp (𝑳 I g)) := by
ext h
  rw [ContMDiffMap.comp_apply, L_apply, ← evalAt_apply, evalAt_mul, hfdifferential_apply,
    fdifferential_apply, evalAt_apply]
```

-- snippet 40
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_injective (L78-L81)
theorem coe_injective :
    @Function.Injective (LeftInvariantDerivation I G) (_ → C^∞⟮I, G; 𝕜⟯) DFunLike.coe :=
DFunLike.coe_injective
```
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Geometry.Manifold.IsManifold.ExtChartAt).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem FiniteDimensional.of_locallyCompact_manifold
    [CompleteSpace 𝕜] (I : ModelWithCorners 𝕜 E H) [Nonempty M] [LocallyCompactSpace M] :
    FiniteDimensional 𝕜 E
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend (L75-L77)
def extend : PartialEquiv M E :=
f.toPartialEquiv ≫ I.toPartialEquiv
```

-- snippet 2
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_coe (L78-L80)
theorem extend_coe : ⇑(f.extend I) = I ∘ f :=
rfl
```

-- snippet 3
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_coe_symm (L81-L83)
theorem extend_coe_symm : ⇑(f.extend I).symm = f.symm ∘ I.symm :=
rfl
```

-- snippet 4
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_source (L84-L86)
theorem extend_source : (f.extend I).source = f.source := by
rw [extend, PartialEquiv.trans_source, I.source_eq, preimage_univ, inter_univ]
```

-- snippet 5
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.isOpen_extend_source (L87-L90)
theorem isOpen_extend_source : IsOpen (f.extend I).source := by
rw [extend_source]
  exact f.open_source
```

-- snippet 6
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_target (L91-L93)
theorem extend_target : (f.extend I).target = I.symm ⁻¹' f.target ∩ range I := by
simp_rw [extend, PartialEquiv.trans_target, I.target_eq, I.toPartialEquiv_coe_symm, inter_comm]
```

-- snippet 7
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_target' (L94-L96)
theorem extend_target' : (f.extend I).target = I '' f.target := by
rw [extend, PartialEquiv.trans_target'', I.source_eq, univ_inter, I.toPartialEquiv_coe]
```

-- snippet 8
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_target_eq_image_source (L97-L99)
theorem extend_target_eq_image_source : (f.extend I).target = (f.extend I) '' f.source := by
rw [f.extend_target', ← f.image_source_eq_target, ← image_comp, f.extend_coe]
```

-- snippet 9
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.isOpen_extend_target (L100-L103)
lemma isOpen_extend_target [I.Boundaryless] : IsOpen (f.extend I).target := by
rw [extend_target, I.range_eq_univ, inter_univ]
  exact I.continuous_symm.isOpen_preimage _ f.open_target
```

-- snippet 10
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.mapsTo_extend (L104-L109)
theorem mapsTo_extend (hs : s ⊆ f.source) :
    MapsTo (f.extend I) s ((f.extend I).symm ⁻¹' s ∩ range I) := by
rw [mapsTo_iff_image_subset, extend_coe, extend_coe_symm, preimage_comp, ← I.image_eq, image_comp,
    f.image_eq_target_inter_inv_preimage hs]
  exact image_mono inter_subset_right
```

-- snippet 11
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_left_inv (L110-L113)
theorem extend_left_inv {x : M} (hxf : x ∈ f.source) : (f.extend I).symm (f.extend I x) = x :=
(f.extend I).left_inv <| by rwa [f.extend_source]

/-- Variant of `f.extend_left_inv I`, stated in terms of images. -/
```

-- snippet 12
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_left_inv' (L114-L116)
lemma extend_left_inv' (ht : t ⊆ f.source) : ((f.extend I).symm ∘ (f.extend I)) '' t = t :=
EqOn.image_eq_self (fun _ hx ↦ f.extend_left_inv (ht hx))
```

-- snippet 13
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_source_mem_nhds (L117-L119)
theorem extend_source_mem_nhds {x : M} (h : x ∈ f.source) : (f.extend I).source ∈ 𝓝 x :=
(isOpen_extend_source f).mem_nhds <| by rwa [f.extend_source]
```

-- snippet 14
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_source_mem_nhdsWithin (L120-L122)
theorem extend_source_mem_nhdsWithin {x : M} (h : x ∈ f.source) : (f.extend I).source ∈ 𝓝[s] x :=
mem_nhdsWithin_of_mem_nhds <| extend_source_mem_nhds f h
```

-- snippet 15
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.continuousOn_extend (L123-L127)
theorem continuousOn_extend : ContinuousOn (f.extend I) (f.extend I).source := by
refine I.continuous.comp_continuousOn ?_
  rw [extend_source]
  exact f.continuousOn
```

-- snippet 16
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.hasNatScalar (L152-L154)
instance hasNatScalar : SMul ℕ (LeftInvariantDerivation I G) where
  smul r X := ⟨r • X.1, fun g => by simp_rw [LinearMap.map_smul_of_tower _ r, left_invariant']⟩
```

-- snippet 17
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_apply (L194-L196)
theorem evalAt_apply : evalAt g X f = (X f) g :=
rfl
```

-- snippet 18
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_zero (L132-L134)
theorem coe_zero : ⇑(0 : LeftInvariantDerivation I G) = 0 :=
rfl
```

-- snippet 19
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_add (L144-L146)
theorem lift_add : (↑(X + Y) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = X + Y :=
rfl
```

-- snippet 20
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.commutator_coe_derivation (L231-L236)
theorem commutator_coe_derivation :
    ⇑⁅X, Y⁆ =
      (⁅(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯), Y⁆ :
        Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) :=
rfl
```

-- snippet 21
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.toFun_eq_coe (L75-L77)
theorem toFun_eq_coe : X.toFun = ⇑X :=
rfl
```

-- snippet 22
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_smul (L172-L177)
theorem lift_smul (k : 𝕜) : (k • X).1 = k • X.1 :=
rfl

variable (I G)

/-- The coercion to function is a monoid homomorphism. -/
```

-- snippet 23
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_mul (L205-L211)
theorem evalAt_mul : evalAt (g * h) X = 𝒅ₕ (L_apply I g h) (evalAt h X) := by
ext f
  rw [← left_invariant, hfdifferential_apply, hfdifferential_apply, L_mul, fdifferential_comp,
    fdifferential_apply]
  simp only [ContMDiffMap.comp_apply, LinearMap.comp_apply]
  rw [fdifferential_apply, ← hfdifferential_apply (smoothLeftMul_one I h), left_invariant]
```

-- snippet 24
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.ext (L83-L86)
theorem ext (h : ∀ f, X f = Y f) : X = Y := DFunLike.ext _ _ h
variable (X Y f)
```

-- snippet 25
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt (L189-L193)
def evalAt : LeftInvariantDerivation I G →ₗ[𝕜] PointDerivation I g where
  toFun X := Derivation.evalAt g X.1
map_add' _ _ := rfl
  map_smul' _ _ := rfl
```

-- snippet 26
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_add (L128-L130)
theorem coe_add : ⇑(X + Y) = X + Y :=
rfl
```

-- snippet 27
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_neg (L136-L138)
theorem coe_neg : ⇑(-X) = -X :=
rfl
```

-- snippet 28
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coeFnAddMonoidHom (L179-L183)
def coeFnAddMonoidHom : LeftInvariantDerivation I G →+ C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯ :=
⟨⟨DFunLike.coe, coe_zero⟩, coe_add⟩

variable {I G}
```

-- snippet 29
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_smul (L168-L170)
theorem coe_smul : ⇑(r • X) = r • ⇑X :=
rfl
```

-- snippet 30
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_coe (L198-L200)
theorem evalAt_coe : Derivation.evalAt g ↑X = evalAt g X :=
rfl
```

-- snippet 31
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.comp_L (L212-L216)
theorem comp_L : (X f).comp (𝑳 I g) = X (f.comp (𝑳 I g)) := by
ext h
  rw [ContMDiffMap.comp_apply, L_apply, ← evalAt_apply, evalAt_mul, hfdifferential_apply,
    fdifferential_apply, evalAt_apply]
```

-- snippet 32
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_derivation (L87-L91)
theorem coe_derivation :
    ⇑(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = (X : C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯) :=
rfl

/-- Premature version of the lemma. Prefer using `left_invariant` instead. -/
```

-- snippet 33
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.left_invariant' (L92-L105)
theorem left_invariant' :
    𝒅ₕ (smoothLeftMul_one I g) (Derivation.evalAt (1 : G) ↑X) = Derivation.evalAt g ↑X :=
left_invariant'' X g

protected theorem map_add : X (f + f') = X f + X f' := by simp

protected theorem map_zero : X 0 = 0 := by simp

protected theorem map_neg : X (-f) = -X f := by simp

protected theorem map_sub : X (f - f') = X f - X f' := by simp

protected theorem map_smul : X (r • f) = r • X f := by simp
```

-- snippet 34
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_zero (L148-L151)
theorem lift_zero :
    (↑(0 : LeftInvariantDerivation I G) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = 0 :=
rfl
```

-- snippet 35
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_sub (L140-L142)
theorem coe_sub : ⇑(X - Y) = X - Y :=
rfl
```

-- snippet 36
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.hasIntScalar (L155-L157)
instance hasIntScalar : SMul ℤ (LeftInvariantDerivation I G) where
  smul r X := ⟨r • X.1, fun g => by simp_rw [LinearMap.map_smul_of_tower _ r, left_invariant']⟩
```

-- snippet 37
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.toDerivation_injective (L61-L64)
theorem toDerivation_injective :
    Function.Injective (toDerivation : LeftInvariantDerivation I G → _) :=
fun X Y h => by cases X; cases Y; congr
```

-- snippet 38
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.leibniz (L107-L109)
theorem leibniz : X (f * f') = f • X f' + f' • X f :=
X.leibniz' _ _
```

-- snippet 39
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.left_invariant (L201-L204)
theorem left_invariant : 𝒅ₕ (smoothLeftMul_one I g) (evalAt (1 : G) X) = evalAt g X :=
X.left_invariant'' g

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 40
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_injective (L78-L81)
theorem coe_injective :
    @Function.Injective (LeftInvariantDerivation I G) (_ → C^∞⟮I, G; 𝕜⟯) DFunLike.coe :=
DFunLike.coe_injective
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Geometry.Manifold.IsManifold.ExtChartAt).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem FiniteDimensional.of_locallyCompact_manifold
    [CompleteSpace 𝕜] (I : ModelWithCorners 𝕜 E H) [Nonempty M] [LocallyCompactSpace M] :
    FiniteDimensional 𝕜 E
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend (L75-L77)
def extend : PartialEquiv M E :=
f.toPartialEquiv ≫ I.toPartialEquiv
```

-- snippet 2
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_coe (L78-L80)
theorem extend_coe : ⇑(f.extend I) = I ∘ f :=
rfl
```

-- snippet 3
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_coe_symm (L81-L83)
theorem extend_coe_symm : ⇑(f.extend I).symm = f.symm ∘ I.symm :=
rfl
```

-- snippet 4
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_source (L84-L86)
theorem extend_source : (f.extend I).source = f.source := by
rw [extend, PartialEquiv.trans_source, I.source_eq, preimage_univ, inter_univ]
```

-- snippet 5
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.isOpen_extend_source (L87-L90)
theorem isOpen_extend_source : IsOpen (f.extend I).source := by
rw [extend_source]
  exact f.open_source
```

-- snippet 6
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_target (L91-L93)
theorem extend_target : (f.extend I).target = I.symm ⁻¹' f.target ∩ range I := by
simp_rw [extend, PartialEquiv.trans_target, I.target_eq, I.toPartialEquiv_coe_symm, inter_comm]
```

-- snippet 7
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_target' (L94-L96)
theorem extend_target' : (f.extend I).target = I '' f.target := by
rw [extend, PartialEquiv.trans_target'', I.source_eq, univ_inter, I.toPartialEquiv_coe]
```

-- snippet 8
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_target_eq_image_source (L97-L99)
theorem extend_target_eq_image_source : (f.extend I).target = (f.extend I) '' f.source := by
rw [f.extend_target', ← f.image_source_eq_target, ← image_comp, f.extend_coe]
```

-- snippet 9
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.isOpen_extend_target (L100-L103)
lemma isOpen_extend_target [I.Boundaryless] : IsOpen (f.extend I).target := by
rw [extend_target, I.range_eq_univ, inter_univ]
  exact I.continuous_symm.isOpen_preimage _ f.open_target
```

-- snippet 10
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.mapsTo_extend (L104-L109)
theorem mapsTo_extend (hs : s ⊆ f.source) :
    MapsTo (f.extend I) s ((f.extend I).symm ⁻¹' s ∩ range I) := by
rw [mapsTo_iff_image_subset, extend_coe, extend_coe_symm, preimage_comp, ← I.image_eq, image_comp,
    f.image_eq_target_inter_inv_preimage hs]
  exact image_mono inter_subset_right
```

-- snippet 11
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_left_inv (L110-L113)
theorem extend_left_inv {x : M} (hxf : x ∈ f.source) : (f.extend I).symm (f.extend I x) = x :=
(f.extend I).left_inv <| by rwa [f.extend_source]

/-- Variant of `f.extend_left_inv I`, stated in terms of images. -/
```

-- snippet 12
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_left_inv' (L114-L116)
lemma extend_left_inv' (ht : t ⊆ f.source) : ((f.extend I).symm ∘ (f.extend I)) '' t = t :=
EqOn.image_eq_self (fun _ hx ↦ f.extend_left_inv (ht hx))
```

-- snippet 13
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_source_mem_nhds (L117-L119)
theorem extend_source_mem_nhds {x : M} (h : x ∈ f.source) : (f.extend I).source ∈ 𝓝 x :=
(isOpen_extend_source f).mem_nhds <| by rwa [f.extend_source]
```

-- snippet 14
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.extend_source_mem_nhdsWithin (L120-L122)
theorem extend_source_mem_nhdsWithin {x : M} (h : x ∈ f.source) : (f.extend I).source ∈ 𝓝[s] x :=
mem_nhdsWithin_of_mem_nhds <| extend_source_mem_nhds f h
```

-- snippet 15
```lean
-- Mathlib.Geometry.Manifold.IsManifold.ExtChartAt.continuousOn_extend (L123-L127)
theorem continuousOn_extend : ContinuousOn (f.extend I) (f.extend I).source := by
refine I.continuous.comp_continuousOn ?_
  rw [extend_source]
  exact f.continuousOn
```

-- snippet 16
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.hasNatScalar (L152-L154)
instance hasNatScalar : SMul ℕ (LeftInvariantDerivation I G) where
  smul r X := ⟨r • X.1, fun g => by simp_rw [LinearMap.map_smul_of_tower _ r, left_invariant']⟩
```

-- snippet 17
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_apply (L194-L196)
theorem evalAt_apply : evalAt g X f = (X f) g :=
rfl
```

-- snippet 18
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_zero (L132-L134)
theorem coe_zero : ⇑(0 : LeftInvariantDerivation I G) = 0 :=
rfl
```

-- snippet 19
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_add (L144-L146)
theorem lift_add : (↑(X + Y) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = X + Y :=
rfl
```

-- snippet 20
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.commutator_coe_derivation (L231-L236)
theorem commutator_coe_derivation :
    ⇑⁅X, Y⁆ =
      (⁅(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯), Y⁆ :
        Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) :=
rfl
```

-- snippet 21
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.toFun_eq_coe (L75-L77)
theorem toFun_eq_coe : X.toFun = ⇑X :=
rfl
```

-- snippet 22
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_smul (L172-L177)
theorem lift_smul (k : 𝕜) : (k • X).1 = k • X.1 :=
rfl

variable (I G)

/-- The coercion to function is a monoid homomorphism. -/
```

-- snippet 23
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_mul (L205-L211)
theorem evalAt_mul : evalAt (g * h) X = 𝒅ₕ (L_apply I g h) (evalAt h X) := by
ext f
  rw [← left_invariant, hfdifferential_apply, hfdifferential_apply, L_mul, fdifferential_comp,
    fdifferential_apply]
  simp only [ContMDiffMap.comp_apply, LinearMap.comp_apply]
  rw [fdifferential_apply, ← hfdifferential_apply (smoothLeftMul_one I h), left_invariant]
```

-- snippet 24
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.ext (L83-L86)
theorem ext (h : ∀ f, X f = Y f) : X = Y := DFunLike.ext _ _ h
variable (X Y f)
```

-- snippet 25
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt (L189-L193)
def evalAt : LeftInvariantDerivation I G →ₗ[𝕜] PointDerivation I g where
  toFun X := Derivation.evalAt g X.1
map_add' _ _ := rfl
  map_smul' _ _ := rfl
```

-- snippet 26
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_add (L128-L130)
theorem coe_add : ⇑(X + Y) = X + Y :=
rfl
```

-- snippet 27
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_neg (L136-L138)
theorem coe_neg : ⇑(-X) = -X :=
rfl
```

-- snippet 28
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coeFnAddMonoidHom (L179-L183)
def coeFnAddMonoidHom : LeftInvariantDerivation I G →+ C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯ :=
⟨⟨DFunLike.coe, coe_zero⟩, coe_add⟩

variable {I G}
```

-- snippet 29
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_smul (L168-L170)
theorem coe_smul : ⇑(r • X) = r • ⇑X :=
rfl
```

-- snippet 30
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.evalAt_coe (L198-L200)
theorem evalAt_coe : Derivation.evalAt g ↑X = evalAt g X :=
rfl
```

-- snippet 31
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.comp_L (L212-L216)
theorem comp_L : (X f).comp (𝑳 I g) = X (f.comp (𝑳 I g)) := by
ext h
  rw [ContMDiffMap.comp_apply, L_apply, ← evalAt_apply, evalAt_mul, hfdifferential_apply,
    fdifferential_apply, evalAt_apply]
```

-- snippet 32
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_derivation (L87-L91)
theorem coe_derivation :
    ⇑(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = (X : C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯) :=
rfl

/-- Premature version of the lemma. Prefer using `left_invariant` instead. -/
```

-- snippet 33
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.left_invariant' (L92-L105)
theorem left_invariant' :
    𝒅ₕ (smoothLeftMul_one I g) (Derivation.evalAt (1 : G) ↑X) = Derivation.evalAt g ↑X :=
left_invariant'' X g

protected theorem map_add : X (f + f') = X f + X f' := by simp

protected theorem map_zero : X 0 = 0 := by simp

protected theorem map_neg : X (-f) = -X f := by simp

protected theorem map_sub : X (f - f') = X f - X f' := by simp

protected theorem map_smul : X (r • f) = r • X f := by simp
```

-- snippet 34
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.lift_zero (L148-L151)
theorem lift_zero :
    (↑(0 : LeftInvariantDerivation I G) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = 0 :=
rfl
```

-- snippet 35
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_sub (L140-L142)
theorem coe_sub : ⇑(X - Y) = X - Y :=
rfl
```

-- snippet 36
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.hasIntScalar (L155-L157)
instance hasIntScalar : SMul ℤ (LeftInvariantDerivation I G) where
  smul r X := ⟨r • X.1, fun g => by simp_rw [LinearMap.map_smul_of_tower _ r, left_invariant']⟩
```

-- snippet 37
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.toDerivation_injective (L61-L64)
theorem toDerivation_injective :
    Function.Injective (toDerivation : LeftInvariantDerivation I G → _) :=
fun X Y h => by cases X; cases Y; congr
```

-- snippet 38
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.leibniz (L107-L109)
theorem leibniz : X (f * f') = f • X f' + f' • X f :=
X.leibniz' _ _
```

-- snippet 39
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.left_invariant (L201-L204)
theorem left_invariant : 𝒅ₕ (smoothLeftMul_one I g) (evalAt (1 : G) X) = evalAt g X :=
X.left_invariant'' g

set_option backward.isDefEq.respectTransparency false in
```

-- snippet 40
```lean
-- Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation.coe_injective (L78-L81)
theorem coe_injective :
    @Function.Injective (LeftInvariantDerivation I G) (_ → C^∞⟮I, G; 𝕜⟯) DFunLike.coe :=
DFunLike.coe_injective
```
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.Constructions.BorelSpace.Metric).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem measurable_norm : Measurable (norm : α → ℝ)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurableSet_ball (L46-L48)
theorem measurableSet_ball : MeasurableSet (Metric.ball x ε) :=
Metric.isOpen_ball.measurableSet
```

-- snippet 2
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurableSet_closedBall (L50-L52)
theorem measurableSet_closedBall : MeasurableSet (Metric.closedBall x ε) :=
Metric.isClosed_closedBall.measurableSet
```

-- snippet 3
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_infDist (L53-L55)
theorem measurable_infDist {s : Set α} : Measurable fun x => infDist x s :=
(continuous_infDist_pt s).measurable
```

-- snippet 4
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.Measurable (L57-L60)
theorem Measurable.infDist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infDist (f x) s :=
measurable_infDist.comp hf
```

-- snippet 5
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_infNndist (L61-L63)
theorem measurable_infNndist {s : Set α} : Measurable fun x => infNndist x s :=
(continuous_infNndist_pt s).measurable
```

-- snippet 6
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.Measurable (L65-L68)
theorem Measurable.infNndist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infNndist (f x) s :=
measurable_infNndist.comp hf
```

-- snippet 7
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_dist (L73-L75)
theorem measurable_dist : Measurable fun p : α × α => dist p.1 p.2 :=
continuous_dist.measurable
```

-- snippet 8
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.Measurable (L77-L80)
theorem Measurable.dist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => dist (f b) (g b) :=
continuous_dist.measurable2 hf hg
```

-- snippet 9
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.AEMeasurable (L82-L86)
lemma AEMeasurable.dist {f g : β → α} {μ : Measure β}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun b ↦ dist (f b) (g b)) μ :=
continuous_dist.aemeasurable2 hf hg
```

-- snippet 10
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_nndist (L87-L89)
theorem measurable_nndist : Measurable fun p : α × α => nndist p.1 p.2 :=
continuous_nndist.measurable
```

-- snippet 11
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.Measurable (L91-L94)
theorem Measurable.nndist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => nndist (f b) (g b) :=
continuous_nndist.measurable2 hf hg
```

-- snippet 12
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurableSet_eball (L107-L109)
theorem measurableSet_eball : MeasurableSet (Metric.eball x ε) :=
Metric.isOpen_eball.measurableSet
```

-- snippet 13
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_edist_right (L111-L112)
theorem measurable_edist_right : Measurable (edist x) := by fun_prop
```

-- snippet 14
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_edist_left (L114-L115)
theorem measurable_edist_left : Measurable fun y ↦ edist y x := by fun_prop
```

-- snippet 15
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_infEDist (L116-L118)
theorem measurable_infEDist {s : Set α} : Measurable fun x => infEDist x s :=
continuous_infEDist.measurable
```

-- snippet 16
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.Countable (L193-L198)
instance Countable.instBorelSpace [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
    [TopologicalSpace α] [DiscreteTopology α] : BorelSpace α := by
have : ∀ s, @MeasurableSet α inferInstance s := fun s ↦ s.to_countable.measurableSet
  have : ∀ s, @MeasurableSet α (borel α) s := fun s ↦ measurableSet_generateFrom (isOpen_discrete s)
  exact ⟨by aesop⟩
```

-- snippet 17
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_anti (L56-L58)
theorem borel_anti : Antitone (@borel α) := fun _ _ h =>
MeasurableSpace.generateFrom_le fun _ hs => .basic _ (h _ hs)
```

-- snippet 18
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_eq_generateFrom_isClosed (L89-L97)
theorem borel_eq_generateFrom_isClosed [TopologicalSpace α] :
    borel α = .generateFrom { s | IsClosed s } :=
le_antisymm
    (generateFrom_le fun _t ht =>
      @MeasurableSet.of_compl α _ (generateFrom { s | IsClosed s })
        (GenerateMeasurable.basic _ <| isClosed_compl_iff.2 ht))
    (generateFrom_le fun _t ht =>
      @MeasurableSet.of_compl α _ (borel α) (GenerateMeasurable.basic _ <| isOpen_compl_iff.2 ht))
```

-- snippet 19
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.IsGδ (L250-L253)
theorem IsGδ.measurableSet (h : IsGδ s) : MeasurableSet s := by
rcases h with ⟨S, hSo, hSc, rfl⟩
  exact MeasurableSet.sInter hSc fun t ht => (hSo t ht).measurableSet
```

-- snippet 20
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_eq_generateFrom_of_subbasis (L62-L77)
theorem borel_eq_generateFrom_of_subbasis {s : Set (Set α)} [t : TopologicalSpace α]
    [SecondCountableTopology α] (hs : t = .generateFrom s) : borel α = .generateFrom s :=
le_antisymm
    (generateFrom_le fun u (hu : t.IsOpen u) => by
      rw [hs] at hu
      induction hu with
      | basic u hu => exact GenerateMeasurable.basic u hu
      | univ => exact @MeasurableSet.univ α (generateFrom s)
      | inter s₁ s₂ _ _ hs₁ hs₂ => exact @MeasurableSet.inter α (generateFrom s) _ _ hs₁ hs₂
      | sUnion f hf ih =>
        rcases isOpen_sUnion_countable f (by rwa [hs]) with ⟨v, hv, vf, vu⟩
        rw [← vu]
        exact @MeasurableSet.sUnion α (generateFrom s) _ hv fun x xv => ih _ (vf xv))
    (generateFrom_le fun u hu =>
      GenerateMeasurable.basic _ <| show t.IsOpen u by rw [hs]; exact GenerateOpen.basic _ hu)
```

-- snippet 21
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel (L53-L55)
def borel (α : Type u) [TopologicalSpace α] : MeasurableSpace α :=
generateFrom { s : Set α | IsOpen s }
```

-- snippet 22
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.IsCompact (L264-L266)
theorem IsCompact.measurableSet [T2Space α] (h : IsCompact s) : MeasurableSet s :=
h.isClosed.measurableSet
```

-- snippet 23
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.measurableSet_of_continuousAt (L254-L257)
theorem measurableSet_of_continuousAt {β} [PseudoEMetricSpace β] (f : α → β) :
    MeasurableSet { x | ContinuousAt f x } :=
(IsGδ.setOf_continuousAt f).measurableSet
```

-- snippet 24
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.Inseparable (L272-L278)
theorem Inseparable.mem_measurableSet_iff {x y : γ} (h : Inseparable x y) {s : Set γ}
    (hs : MeasurableSet s) : x ∈ s ↔ y ∈ s :=
MeasurableSet.induction_on_open (fun _ ↦ h.mem_open_iff) (fun _ _ ↦ Iff.not)
    (fun _ _ _ h ↦ by simp [h]) s hs

/-- If `K` is a compact set in an R₁ space and `s ⊇ K` is a Borel measurable superset,
then `s` includes the closure of `K` as well. -/
```

-- snippet 25
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_eq_top_of_discrete (L59-L61)
theorem borel_eq_top_of_discrete [TopologicalSpace α] [DiscreteTopology α] : borel α = ⊤ :=
top_le_iff.1 fun s _ => GenerateMeasurable.basic s (isOpen_discrete s)
```

-- snippet 26
```lean
-- Mathlib.MeasureTheory.Constructions.AddChar.instDiscreteMeasurableSpace (L32-L36)
instance instDiscreteMeasurableSpace [DiscreteMeasurableSpace A] [Finite A] :
    DiscreteMeasurableSpace (AddChar A M) :=
⟨fun _ ↦ trivial⟩

end AddChar
```

-- snippet 27
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.IsClosed (L258-L260)
theorem IsClosed.measurableSet (h : IsClosed s) : MeasurableSet s :=
h.isOpen_compl.measurableSet.of_compl
```

-- snippet 28
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.isPiSystem_isClosed (L86-L88)
lemma isPiSystem_isClosed [TopologicalSpace α] : IsPiSystem ({s : Set α | IsClosed s}) :=
fun _s hs _t ht _ ↦ IsClosed.inter hs ht
```

-- snippet 29
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.Subtype (L189-L192)
instance Subtype.borelSpace {α : Type*} [TopologicalSpace α] [MeasurableSpace α]
    [hα : BorelSpace α] (p : α → Prop) : BorelSpace (Subtype p) :=
⟨by borelize α; symm; apply borel_comap⟩
```

-- snippet 30
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.TopologicalSpace (L78-L82)
theorem TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom [TopologicalSpace α]
    [SecondCountableTopology α] {s : Set (Set α)} (hs : IsTopologicalBasis s) :
    borel α = .generateFrom s :=
borel_eq_generateFrom_of_subbasis hs.eq_generateFrom
```

-- snippet 31
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.measurableSet_frontier (L303-L305)
theorem measurableSet_frontier : MeasurableSet (frontier s) :=
measurableSet_closure.diff measurableSet_interior
```

-- snippet 32
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_comap (L98-L101)
theorem borel_comap {f : α → β} {t : TopologicalSpace β} :
    @borel α (t.induced f) = (@borel β t).comap f :=
comap_generateFrom.symm
```

-- snippet 33
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.measurableSet_interior (L247-L249)
theorem measurableSet_interior : MeasurableSet (interior s) :=
isOpen_interior.measurableSet
```

-- snippet 34
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.isPiSystem_isOpen (L83-L85)
theorem isPiSystem_isOpen [TopologicalSpace α] : IsPiSystem ({s : Set α | IsOpen s}) :=
fun _s hs _t ht _ => IsOpen.inter hs ht
```

-- snippet 35
```lean
-- Mathlib.MeasureTheory.Constructions.AddChar.instMeasurableSpace (L28-L31)
instance instMeasurableSpace [DiscreteMeasurableSpace A] [Finite A] :
    MeasurableSpace (AddChar A M) :=
⊤
```

-- snippet 36
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.measurableSet_closure (L299-L301)
theorem measurableSet_closure : MeasurableSet (closure s) :=
isClosed_closure.measurableSet
```

-- snippet 37
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.opensMeasurableSpace_iff_forall_measurableSet (L205-L210)
lemma opensMeasurableSpace_iff_forall_measurableSet
    [TopologicalSpace α] [MeasurableSpace α] :
    OpensMeasurableSpace α ↔ (∀ (s : Set α), IsOpen s → MeasurableSet s) := by
refine ⟨fun h s hs ↦ ?_, fun h ↦ ⟨generateFrom_le h⟩⟩
  exact OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ hs
```

-- snippet 38
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.MeasurableSet (L232-L240)
theorem MeasurableSet.induction_on_open {C : ∀ s : Set γ, MeasurableSet s → Prop}
    (isOpen : ∀ U (hU : IsOpen U), C U hU.measurableSet)
    (compl : ∀ t (ht : MeasurableSet t), C t ht → C tᶜ ht.compl)
    (iUnion : ∀ f : ℕ → Set γ, Pairwise (Disjoint on f) → ∀ (hf : ∀ i, MeasurableSet (f i)),
      (∀ i, C (f i) (hf i)) → C (⋃ i, f i) (.iUnion hf)) :
    ∀ t (ht : MeasurableSet t), C t ht := fun t ht ↦
MeasurableSpace.induction_on_inter BorelSpace.measurable_eq isPiSystem_isOpen
    (isOpen _ isOpen_empty) isOpen compl iUnion t ht
```

-- snippet 39
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.IsOpen (L224-L226)
theorem IsOpen.measurableSet (h : IsOpen s) : MeasurableSet s :=
OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ h
```

-- snippet 40
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.Continuous (L102-L175)
theorem Continuous.borel_measurable [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
    (hf : Continuous f) : @Measurable α β (borel α) (borel β) f :=
Measurable.of_le_map <|
    generateFrom_le fun s hs => GenerateMeasurable.basic (f ⁻¹' s) (hs.preimage hf)

/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that
all open sets are measurable. -/
class OpensMeasurableSpace (α : Type*) [TopologicalSpace α] [h : MeasurableSpace α] : Prop where
  /-- Borel-measurable sets are measurable. -/
  borel_le : borel α ≤ h

/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that
the `σ`-algebra of measurable sets is exactly the `σ`-algebra generated by open sets. -/
class BorelSpace (α : Type*) [TopologicalSpace α] [MeasurableSpace α] : Prop where
  /-- The measurable sets are exactly the Borel-measurable sets. -/
  measurable_eq : ‹MeasurableSpace α› = borel α

namespace Mathlib.Tactic.Borelize

open Lean Elab Term Tactic Meta

/-- The behaviour of `borelize α` depends on the existing assumptions on `α`.

- if `α` is a topological space with instances `[MeasurableSpace α] [BorelSpace α]`, then
  `borelize α` replaces the former instance by `borel α`;
- otherwise, `borelize α` adds instances `borel α : MeasurableSpace α` and `⟨rfl⟩ : BorelSpace α`.

Finally, `borelize α β γ` runs `borelize α; borelize β; borelize γ`.
-/
syntax "borelize" (ppSpace colGt term:max)* : tactic

/-- Add instances `borel e : MeasurableSpace e` and `⟨rfl⟩ : BorelSpace e`. -/
meta def addBorelInstance (e : Expr) : TacticM Unit := do
  let t ← Lean.Elab.Term.exprToSyntax e
  evalTactic <| ← `(tactic|
    refine_lift
      letI : MeasurableSpace $t := borel $t
      haveI : BorelSpace $t := ⟨rfl⟩
      ?_)

/-- Given a type `e`, an assumption `i : MeasurableSpace e`, and an instance `[BorelSpace e]`,
replace `i` with `borel e`. -/
meta def borelToRefl (e : Expr) (i : FVarId) : TacticM Unit := do
  let te ← Lean.Elab.Term.exprToSyntax e
  evalTactic <| ← `(tactic|
    have := @BorelSpace.measurable_eq $te _ _ _)
  try
    liftMetaTactic fun m => return [← subst m i]
  catch _ =>
    let et ← synthInstance (← mkAppOptM ``TopologicalSpace #[e])
    throwError m!"\
      `‹TopologicalSpace {e}› := {et}\n\
      depends on\n\
      {Expr.fvar i} : MeasurableSpace {e}`\n\
      so `borelize` isn't available"
  evalTactic <| ← `(tactic|
    refine_lift
      letI : MeasurableSpace $te := borel $te
      ?_)

/-- Given a type `$t`, if there is an assumption `[i : MeasurableSpace $t]`, then try to prove
`[BorelSpace $t]` and replace `i` with `borel $t`. Otherwise, add instances
`borel $t : MeasurableSpace $t` and `⟨rfl⟩ : BorelSpace $t`. -/
meta def borelize (t : Term) : TacticM Unit := withMainContext do
  let u ← mkFreshLevelMVar
  let e ← withoutRecover <| Tactic.elabTermEnsuringType t (mkSort (mkLevelSucc u))
  let i? ← findLocalDeclWithType? (← mkAppOptM ``MeasurableSpace #[e])
  i?.elim (addBorelInstance e) (borelToRefl e)

elab_rules : tactic
  | `(tactic| borelize $[$t:term]*) => t.forM borelize

end Mathlib.Tactic.Borelize
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.MeasureTheory.Constructions.BorelSpace.Metric).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem measurable_norm : Measurable (norm : α → ℝ)
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurableSet_ball (L46-L48)
theorem measurableSet_ball : MeasurableSet (Metric.ball x ε) :=
Metric.isOpen_ball.measurableSet
```

-- snippet 2
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurableSet_closedBall (L50-L52)
theorem measurableSet_closedBall : MeasurableSet (Metric.closedBall x ε) :=
Metric.isClosed_closedBall.measurableSet
```

-- snippet 3
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_infDist (L53-L55)
theorem measurable_infDist {s : Set α} : Measurable fun x => infDist x s :=
(continuous_infDist_pt s).measurable
```

-- snippet 4
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.Measurable (L57-L60)
theorem Measurable.infDist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infDist (f x) s :=
measurable_infDist.comp hf
```

-- snippet 5
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_infNndist (L61-L63)
theorem measurable_infNndist {s : Set α} : Measurable fun x => infNndist x s :=
(continuous_infNndist_pt s).measurable
```

-- snippet 6
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.Measurable (L65-L68)
theorem Measurable.infNndist {f : β → α} (hf : Measurable f) {s : Set α} :
    Measurable fun x => infNndist (f x) s :=
measurable_infNndist.comp hf
```

-- snippet 7
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_dist (L73-L75)
theorem measurable_dist : Measurable fun p : α × α => dist p.1 p.2 :=
continuous_dist.measurable
```

-- snippet 8
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.Measurable (L77-L80)
theorem Measurable.dist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => dist (f b) (g b) :=
continuous_dist.measurable2 hf hg
```

-- snippet 9
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.AEMeasurable (L82-L86)
lemma AEMeasurable.dist {f g : β → α} {μ : Measure β}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun b ↦ dist (f b) (g b)) μ :=
continuous_dist.aemeasurable2 hf hg
```

-- snippet 10
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_nndist (L87-L89)
theorem measurable_nndist : Measurable fun p : α × α => nndist p.1 p.2 :=
continuous_nndist.measurable
```

-- snippet 11
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.Measurable (L91-L94)
theorem Measurable.nndist {f g : β → α} (hf : Measurable f) (hg : Measurable g) :
    Measurable fun b => nndist (f b) (g b) :=
continuous_nndist.measurable2 hf hg
```

-- snippet 12
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurableSet_eball (L107-L109)
theorem measurableSet_eball : MeasurableSet (Metric.eball x ε) :=
Metric.isOpen_eball.measurableSet
```

-- snippet 13
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_edist_right (L111-L112)
theorem measurable_edist_right : Measurable (edist x) := by fun_prop
```

-- snippet 14
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_edist_left (L114-L115)
theorem measurable_edist_left : Measurable fun y ↦ edist y x := by fun_prop
```

-- snippet 15
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Metric.measurable_infEDist (L116-L118)
theorem measurable_infEDist {s : Set α} : Measurable fun x => infEDist x s :=
continuous_infEDist.measurable
```

-- snippet 16
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.Countable (L193-L198)
instance Countable.instBorelSpace [Countable α] [MeasurableSpace α] [MeasurableSingletonClass α]
    [TopologicalSpace α] [DiscreteTopology α] : BorelSpace α := by
have : ∀ s, @MeasurableSet α inferInstance s := fun s ↦ s.to_countable.measurableSet
  have : ∀ s, @MeasurableSet α (borel α) s := fun s ↦ measurableSet_generateFrom (isOpen_discrete s)
  exact ⟨by aesop⟩
```

-- snippet 17
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_anti (L56-L58)
theorem borel_anti : Antitone (@borel α) := fun _ _ h =>
MeasurableSpace.generateFrom_le fun _ hs => .basic _ (h _ hs)
```

-- snippet 18
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_eq_generateFrom_isClosed (L89-L97)
theorem borel_eq_generateFrom_isClosed [TopologicalSpace α] :
    borel α = .generateFrom { s | IsClosed s } :=
le_antisymm
    (generateFrom_le fun _t ht =>
      @MeasurableSet.of_compl α _ (generateFrom { s | IsClosed s })
        (GenerateMeasurable.basic _ <| isClosed_compl_iff.2 ht))
    (generateFrom_le fun _t ht =>
      @MeasurableSet.of_compl α _ (borel α) (GenerateMeasurable.basic _ <| isOpen_compl_iff.2 ht))
```

-- snippet 19
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.IsGδ (L250-L253)
theorem IsGδ.measurableSet (h : IsGδ s) : MeasurableSet s := by
rcases h with ⟨S, hSo, hSc, rfl⟩
  exact MeasurableSet.sInter hSc fun t ht => (hSo t ht).measurableSet
```

-- snippet 20
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_eq_generateFrom_of_subbasis (L62-L77)
theorem borel_eq_generateFrom_of_subbasis {s : Set (Set α)} [t : TopologicalSpace α]
    [SecondCountableTopology α] (hs : t = .generateFrom s) : borel α = .generateFrom s :=
le_antisymm
    (generateFrom_le fun u (hu : t.IsOpen u) => by
      rw [hs] at hu
      induction hu with
      | basic u hu => exact GenerateMeasurable.basic u hu
      | univ => exact @MeasurableSet.univ α (generateFrom s)
      | inter s₁ s₂ _ _ hs₁ hs₂ => exact @MeasurableSet.inter α (generateFrom s) _ _ hs₁ hs₂
      | sUnion f hf ih =>
        rcases isOpen_sUnion_countable f (by rwa [hs]) with ⟨v, hv, vf, vu⟩
        rw [← vu]
        exact @MeasurableSet.sUnion α (generateFrom s) _ hv fun x xv => ih _ (vf xv))
    (generateFrom_le fun u hu =>
      GenerateMeasurable.basic _ <| show t.IsOpen u by rw [hs]; exact GenerateOpen.basic _ hu)
```

-- snippet 21
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel (L53-L55)
def borel (α : Type u) [TopologicalSpace α] : MeasurableSpace α :=
generateFrom { s : Set α | IsOpen s }
```

-- snippet 22
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.IsCompact (L264-L266)
theorem IsCompact.measurableSet [T2Space α] (h : IsCompact s) : MeasurableSet s :=
h.isClosed.measurableSet
```

-- snippet 23
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.measurableSet_of_continuousAt (L254-L257)
theorem measurableSet_of_continuousAt {β} [PseudoEMetricSpace β] (f : α → β) :
    MeasurableSet { x | ContinuousAt f x } :=
(IsGδ.setOf_continuousAt f).measurableSet
```

-- snippet 24
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.Inseparable (L272-L278)
theorem Inseparable.mem_measurableSet_iff {x y : γ} (h : Inseparable x y) {s : Set γ}
    (hs : MeasurableSet s) : x ∈ s ↔ y ∈ s :=
MeasurableSet.induction_on_open (fun _ ↦ h.mem_open_iff) (fun _ _ ↦ Iff.not)
    (fun _ _ _ h ↦ by simp [h]) s hs

/-- If `K` is a compact set in an R₁ space and `s ⊇ K` is a Borel measurable superset,
then `s` includes the closure of `K` as well. -/
```

-- snippet 25
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_eq_top_of_discrete (L59-L61)
theorem borel_eq_top_of_discrete [TopologicalSpace α] [DiscreteTopology α] : borel α = ⊤ :=
top_le_iff.1 fun s _ => GenerateMeasurable.basic s (isOpen_discrete s)
```

-- snippet 26
```lean
-- Mathlib.MeasureTheory.Constructions.AddChar.instDiscreteMeasurableSpace (L32-L36)
instance instDiscreteMeasurableSpace [DiscreteMeasurableSpace A] [Finite A] :
    DiscreteMeasurableSpace (AddChar A M) :=
⟨fun _ ↦ trivial⟩

end AddChar
```

-- snippet 27
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.IsClosed (L258-L260)
theorem IsClosed.measurableSet (h : IsClosed s) : MeasurableSet s :=
h.isOpen_compl.measurableSet.of_compl
```

-- snippet 28
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.isPiSystem_isClosed (L86-L88)
lemma isPiSystem_isClosed [TopologicalSpace α] : IsPiSystem ({s : Set α | IsClosed s}) :=
fun _s hs _t ht _ ↦ IsClosed.inter hs ht
```

-- snippet 29
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.Subtype (L189-L192)
instance Subtype.borelSpace {α : Type*} [TopologicalSpace α] [MeasurableSpace α]
    [hα : BorelSpace α] (p : α → Prop) : BorelSpace (Subtype p) :=
⟨by borelize α; symm; apply borel_comap⟩
```

-- snippet 30
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.TopologicalSpace (L78-L82)
theorem TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom [TopologicalSpace α]
    [SecondCountableTopology α] {s : Set (Set α)} (hs : IsTopologicalBasis s) :
    borel α = .generateFrom s :=
borel_eq_generateFrom_of_subbasis hs.eq_generateFrom
```

-- snippet 31
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.measurableSet_frontier (L303-L305)
theorem measurableSet_frontier : MeasurableSet (frontier s) :=
measurableSet_closure.diff measurableSet_interior
```

-- snippet 32
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.borel_comap (L98-L101)
theorem borel_comap {f : α → β} {t : TopologicalSpace β} :
    @borel α (t.induced f) = (@borel β t).comap f :=
comap_generateFrom.symm
```

-- snippet 33
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.measurableSet_interior (L247-L249)
theorem measurableSet_interior : MeasurableSet (interior s) :=
isOpen_interior.measurableSet
```

-- snippet 34
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.isPiSystem_isOpen (L83-L85)
theorem isPiSystem_isOpen [TopologicalSpace α] : IsPiSystem ({s : Set α | IsOpen s}) :=
fun _s hs _t ht _ => IsOpen.inter hs ht
```

-- snippet 35
```lean
-- Mathlib.MeasureTheory.Constructions.AddChar.instMeasurableSpace (L28-L31)
instance instMeasurableSpace [DiscreteMeasurableSpace A] [Finite A] :
    MeasurableSpace (AddChar A M) :=
⊤
```

-- snippet 36
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.measurableSet_closure (L299-L301)
theorem measurableSet_closure : MeasurableSet (closure s) :=
isClosed_closure.measurableSet
```

-- snippet 37
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.opensMeasurableSpace_iff_forall_measurableSet (L205-L210)
lemma opensMeasurableSpace_iff_forall_measurableSet
    [TopologicalSpace α] [MeasurableSpace α] :
    OpensMeasurableSpace α ↔ (∀ (s : Set α), IsOpen s → MeasurableSet s) := by
refine ⟨fun h s hs ↦ ?_, fun h ↦ ⟨generateFrom_le h⟩⟩
  exact OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ hs
```

-- snippet 38
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.MeasurableSet (L232-L240)
theorem MeasurableSet.induction_on_open {C : ∀ s : Set γ, MeasurableSet s → Prop}
    (isOpen : ∀ U (hU : IsOpen U), C U hU.measurableSet)
    (compl : ∀ t (ht : MeasurableSet t), C t ht → C tᶜ ht.compl)
    (iUnion : ∀ f : ℕ → Set γ, Pairwise (Disjoint on f) → ∀ (hf : ∀ i, MeasurableSet (f i)),
      (∀ i, C (f i) (hf i)) → C (⋃ i, f i) (.iUnion hf)) :
    ∀ t (ht : MeasurableSet t), C t ht := fun t ht ↦
MeasurableSpace.induction_on_inter BorelSpace.measurable_eq isPiSystem_isOpen
    (isOpen _ isOpen_empty) isOpen compl iUnion t ht
```

-- snippet 39
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.IsOpen (L224-L226)
theorem IsOpen.measurableSet (h : IsOpen s) : MeasurableSet s :=
OpensMeasurableSpace.borel_le _ <| GenerateMeasurable.basic _ h
```

-- snippet 40
```lean
-- Mathlib.MeasureTheory.Constructions.BorelSpace.Basic.Continuous (L102-L175)
theorem Continuous.borel_measurable [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
    (hf : Continuous f) : @Measurable α β (borel α) (borel β) f :=
Measurable.of_le_map <|
    generateFrom_le fun s hs => GenerateMeasurable.basic (f ⁻¹' s) (hs.preimage hf)

/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that
all open sets are measurable. -/
class OpensMeasurableSpace (α : Type*) [TopologicalSpace α] [h : MeasurableSpace α] : Prop where
  /-- Borel-measurable sets are measurable. -/
  borel_le : borel α ≤ h

/-- A space with `MeasurableSpace` and `TopologicalSpace` structures such that
the `σ`-algebra of measurable sets is exactly the `σ`-algebra generated by open sets. -/
class BorelSpace (α : Type*) [TopologicalSpace α] [MeasurableSpace α] : Prop where
  /-- The measurable sets are exactly the Borel-measurable sets. -/
  measurable_eq : ‹MeasurableSpace α› = borel α

namespace Mathlib.Tactic.Borelize

open Lean Elab Term Tactic Meta

/-- The behaviour of `borelize α` depends on the existing assumptions on `α`.

- if `α` is a topological space with instances `[MeasurableSpace α] [BorelSpace α]`, then
  `borelize α` replaces the former instance by `borel α`;
- otherwise, `borelize α` adds instances `borel α : MeasurableSpace α` and `⟨rfl⟩ : BorelSpace α`.

Finally, `borelize α β γ` runs `borelize α; borelize β; borelize γ`.
-/
syntax "borelize" (ppSpace colGt term:max)* : tactic

/-- Add instances `borel e : MeasurableSpace e` and `⟨rfl⟩ : BorelSpace e`. -/
meta def addBorelInstance (e : Expr) : TacticM Unit := do
  let t ← Lean.Elab.Term.exprToSyntax e
  evalTactic <| ← `(tactic|
    refine_lift
      letI : MeasurableSpace $t := borel $t
      haveI : BorelSpace $t := ⟨rfl⟩
      ?_)

/-- Given a type `e`, an assumption `i : MeasurableSpace e`, and an instance `[BorelSpace e]`,
replace `i` with `borel e`. -/
meta def borelToRefl (e : Expr) (i : FVarId) : TacticM Unit := do
  let te ← Lean.Elab.Term.exprToSyntax e
  evalTactic <| ← `(tactic|
    have := @BorelSpace.measurable_eq $te _ _ _)
  try
    liftMetaTactic fun m => return [← subst m i]
  catch _ =>
    let et ← synthInstance (← mkAppOptM ``TopologicalSpace #[e])
    throwError m!"\
      `‹TopologicalSpace {e}› := {et}\n\
      depends on\n\
      {Expr.fvar i} : MeasurableSpace {e}`\n\
      so `borelize` isn't available"
  evalTactic <| ← `(tactic|
    refine_lift
      letI : MeasurableSpace $te := borel $te
      ?_)

/-- Given a type `$t`, if there is an assumption `[i : MeasurableSpace $t]`, then try to prove
`[BorelSpace $t]` and replace `i` with `borel $t`. Otherwise, add instances
`borel $t : MeasurableSpace $t` and `⟨rfl⟩ : BorelSpace $t`. -/
meta def borelize (t : Term) : TacticM Unit := withMainContext do
  let u ← mkFreshLevelMVar
  let e ← withoutRecover <| Tactic.elabTermEnsuringType t (mkSort (mkLevelSucc u))
  let i? ← findLocalDeclWithType? (← mkAppOptM ``MeasurableSpace #[e])
  i?.elim (addBorelInstance e) (borelToRefl e)

elab_rules : tactic
  | `(tactic| borelize $[$t:term]*) => t.forM borelize

end Mathlib.Tactic.Borelize
```
[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.ModelTheory.Substructures).

[User]
Prove the following theorem. The retrieved context below may help.

```lean
theorem comap_le_comap_iff_of_surjective {S T : L.Substructure N} : S.comap f ≤ T.comap f ↔ S ≤ T
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.ModelTheory.Substructures.ClosedUnder (L66-L70)
def ClosedUnder : Prop :=
∀ x : Fin n → M, (∀ i : Fin n, x i ∈ s) → funMap f x ∈ s

variable (L)
```

-- snippet 2
```lean
-- Mathlib.ModelTheory.Substructures.closedUnder_univ (L72-L77)
theorem closedUnder_univ : ClosedUnder f (univ : Set M) := fun _ _ => mem_univ _
variable {L f s} {t : Set M}

namespace ClosedUnder
```

-- snippet 3
```lean
-- Mathlib.ModelTheory.Substructures.inter (L78-L80)
theorem inter (hs : ClosedUnder f s) (ht : ClosedUnder f t) : ClosedUnder f (s ∩ t) := fun x h =>
mem_inter (hs x fun i => mem_of_mem_inter_left (h i)) (ht x fun i => mem_of_mem_inter_right (h i))
```

-- snippet 4
```lean
-- Mathlib.ModelTheory.Substructures.inf (L81-L85)
theorem inf (hs : ClosedUnder f s) (ht : ClosedUnder f t) : ClosedUnder f (s ⊓ t) :=
hs.inter ht

variable {S : Set (Set M)}
```

-- snippet 5
```lean
-- Mathlib.ModelTheory.Substructures.sInf (L86-L104)
theorem sInf (hS : ∀ s, s ∈ S → ClosedUnder f s) : ClosedUnder f (sInf S) := fun x h s hs =>
hS s hs x fun i => h i s hs

end ClosedUnder

end ClosedUnder

variable (L) (M)

/-- A substructure of a structure `M` is a set closed under application of function symbols. -/
structure Substructure where
  /-- The underlying set of this substructure -/
  carrier : Set M
  fun_mem : ∀ {n}, ∀ f : L.Functions n, ClosedUnder f carrier

variable {L} {M}

namespace Substructure
```

-- snippet 6
```lean
-- Mathlib.ModelTheory.Substructures.instSetLike (L107-L109)
instance instSetLike : SetLike (L.Substructure M) M :=
⟨Substructure.carrier, fun p q h => by cases p; cases q; congr⟩
```

-- snippet 7
```lean
-- Mathlib.ModelTheory.Substructures.Simps (L113-L117)
def Simps.coe (S : L.Substructure M) : Set M :=
S

initialize_simps_projections Substructure (carrier → coe, as_prefix coe)
```

-- snippet 8
```lean
-- Mathlib.ModelTheory.Substructures.mem_carrier (L119-L122)
theorem mem_carrier {s : L.Substructure M} {x : M} : x ∈ s.carrier ↔ x ∈ s :=
Iff.rfl

/-- Two substructures are equal if they have the same elements. -/
```

-- snippet 9
```lean
-- Mathlib.ModelTheory.Substructures.ext (L124-L135)
theorem ext {S T : L.Substructure M} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=
SetLike.ext h

/-- Copy a substructure replacing `carrier` with a set that is equal to it. -/
protected def copy (S : L.Substructure M) (s : Set M) (hs : s = S) : L.Substructure M where
  carrier := s
  fun_mem _ f := hs.symm ▸ S.fun_mem _ f

end Substructure

variable {S : L.Substructure M}
```

-- snippet 10
```lean
-- Mathlib.ModelTheory.Substructures.Term (L136-L143)
theorem Term.realize_mem {α : Type*} (t : L.Term α) (xs : α → M) (h : ∀ a, xs a ∈ S) :
    t.realize xs ∈ S := by
induction t with
  | var a => exact h a
  | func f ts ih => exact Substructure.fun_mem _ _ _ ih

namespace Substructure
```

-- snippet 11
```lean
-- Mathlib.ModelTheory.Substructures.coe_copy (L145-L147)
theorem coe_copy {s : Set M} (hs : s = S) : (S.copy s hs : Set M) = s :=
rfl
```

-- snippet 12
```lean
-- Mathlib.ModelTheory.Substructures.copy_eq (L148-L150)
theorem copy_eq {s : Set M} (hs : s = S) : S.copy s hs = S :=
SetLike.coe_injective hs
```

-- snippet 13
```lean
-- Mathlib.ModelTheory.Substructures.constants_mem (L151-L154)
theorem constants_mem (c : L.Constants) : (c : M) ∈ S :=
mem_carrier.2 (S.fun_mem c _ finZeroElim)

/-- The substructure `M` of the structure `M`. -/
```

-- snippet 14
```lean
-- Mathlib.ModelTheory.Substructures.instTop (L155-L158)
instance instTop : Top (L.Substructure M) :=
⟨{  carrier := Set.univ
      fun_mem := fun {_} _ _ _ => Set.mem_univ _ }⟩
```

-- snippet 15
```lean
-- Mathlib.ModelTheory.Substructures.instInhabited (L159-L161)
instance instInhabited : Inhabited (L.Substructure M) :=
⟨⊤⟩
```

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.ModelTheory.Substructures).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]Prove the following theorem. The retrieved context below may help.

```lean
theorem comap_le_comap_iff_of_surjective {S T : L.Substructure N} : S.comap f ≤ T.comap f ↔ S ≤ T
```

## Retrieved context
-- snippet 1
```lean
-- Mathlib.ModelTheory.Substructures.ClosedUnder (L66-L70)
def ClosedUnder : Prop :=
∀ x : Fin n → M, (∀ i : Fin n, x i ∈ s) → funMap f x ∈ s

variable (L)
```

-- snippet 2
```lean
-- Mathlib.ModelTheory.Substructures.closedUnder_univ (L72-L77)
theorem closedUnder_univ : ClosedUnder f (univ : Set M) := fun _ _ => mem_univ _
variable {L f s} {t : Set M}

namespace ClosedUnder
```

-- snippet 3
```lean
-- Mathlib.ModelTheory.Substructures.inter (L78-L80)
theorem inter (hs : ClosedUnder f s) (ht : ClosedUnder f t) : ClosedUnder f (s ∩ t) := fun x h =>
mem_inter (hs x fun i => mem_of_mem_inter_left (h i)) (ht x fun i => mem_of_mem_inter_right (h i))
```

-- snippet 4
```lean
-- Mathlib.ModelTheory.Substructures.inf (L81-L85)
theorem inf (hs : ClosedUnder f s) (ht : ClosedUnder f t) : ClosedUnder f (s ⊓ t) :=
hs.inter ht

variable {S : Set (Set M)}
```

-- snippet 5
```lean
-- Mathlib.ModelTheory.Substructures.sInf (L86-L104)
theorem sInf (hS : ∀ s, s ∈ S → ClosedUnder f s) : ClosedUnder f (sInf S) := fun x h s hs =>
hS s hs x fun i => h i s hs

end ClosedUnder

end ClosedUnder

variable (L) (M)

/-- A substructure of a structure `M` is a set closed under application of function symbols. -/
structure Substructure where
  /-- The underlying set of this substructure -/
  carrier : Set M
  fun_mem : ∀ {n}, ∀ f : L.Functions n, ClosedUnder f carrier

variable {L} {M}

namespace Substructure
```

-- snippet 6
```lean
-- Mathlib.ModelTheory.Substructures.instSetLike (L107-L109)
instance instSetLike : SetLike (L.Substructure M) M :=
⟨Substructure.carrier, fun p q h => by cases p; cases q; congr⟩
```

-- snippet 7
```lean
-- Mathlib.ModelTheory.Substructures.Simps (L113-L117)
def Simps.coe (S : L.Substructure M) : Set M :=
S

initialize_simps_projections Substructure (carrier → coe, as_prefix coe)
```

-- snippet 8
```lean
-- Mathlib.ModelTheory.Substructures.mem_carrier (L119-L122)
theorem mem_carrier {s : L.Substructure M} {x : M} : x ∈ s.carrier ↔ x ∈ s :=
Iff.rfl

/-- Two substructures are equal if they have the same elements. -/
```

-- snippet 9
```lean
-- Mathlib.ModelTheory.Substructures.ext (L124-L135)
theorem ext {S T : L.Substructure M} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=
SetLike.ext h

/-- Copy a substructure replacing `carrier` with a set that is equal to it. -/
protected def copy (S : L.Substructure M) (s : Set M) (hs : s = S) : L.Substructure M where
  carrier := s
  fun_mem _ f := hs.symm ▸ S.fun_mem _ f

end Substructure

variable {S : L.Substructure M}
```

-- snippet 10
```lean
-- Mathlib.ModelTheory.Substructures.Term (L136-L143)
theorem Term.realize_mem {α : Type*} (t : L.Term α) (xs : α → M) (h : ∀ a, xs a ∈ S) :
    t.realize xs ∈ S := by
induction t with
  | var a => exact h a
  | func f ts ih => exact Substructure.fun_mem _ _ _ ih

namespace Substructure
```

-- snippet 11
```lean
-- Mathlib.ModelTheory.Substructures.coe_copy (L145-L147)
theorem coe_copy {s : Set M} (hs : s = S) : (S.copy s hs : Set M) = s :=
rfl
```

-- snippet 12
```lean
-- Mathlib.ModelTheory.Substructures.copy_eq (L148-L150)
theorem copy_eq {s : Set M} (hs : s = S) : S.copy s hs = S :=
SetLike.coe_injective hs
```

-- snippet 13
```lean
-- Mathlib.ModelTheory.Substructures.constants_mem (L151-L154)
theorem constants_mem (c : L.Constants) : (c : M) ∈ S :=
mem_carrier.2 (S.fun_mem c _ finZeroElim)

/-- The substructure `M` of the structure `M`. -/
```

-- snippet 14
```lean
-- Mathlib.ModelTheory.Substructures.instTop (L155-L158)
instance instTop : Top (L.Substructure M) :=
⟨{  carrier := Set.univ
      fun_mem := fun {_} _ _ _ => Set.mem_univ _ }⟩
```

-- snippet 15
```lean
-- Mathlib.ModelTheory.Substructures.instInhabited (L159-L161)
instance instInhabited : Inhabited (L.Substructure M) :=
⟨⊤⟩
```
[/INST]