ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsLocallyClosed.isOpen_preimage_val_closure
lemma IsLocallyClosed.isOpen_preimage_val_closure (hs : IsLocallyClosed s) :
    IsOpen (closure s ↓∩ s) 
Arguments: {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsLocallyClosed s)
Goal: IsOpen (closure s ↓∩ s)
-- ultrafilterBasis_is_basis
theorem ultrafilterBasis_is_basis : TopologicalSpace.IsTopologicalBasis (ultrafilterBasis α) 
Arguments: {α : Type u}
Goal: TopologicalSpace.IsTopologicalBasis (ultrafilterBasis α)
-- ultrafilter_isOpen_basic
theorem ultrafilter_isOpen_basic (s : Set α) : IsOpen { u : Ultrafilter α | s ∈ u } 
Arguments: {α : Type u} (s : Set α)
Goal: IsOpen {u | s ∈ u}
-- ultrafilter_isClosed_basic
theorem ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } 
Arguments: {α : Type u} (s : Set α)
Goal: IsClosed {u | s ∈ u}
-- ultrafilter_converges_iff
theorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :
    ↑u ≤ 𝓝 x ↔ x = joinM u 
Arguments: {α : Type u} {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α}
Goal: ↑u ≤ 𝓝 x ↔ x = joinM u
-- Ultrafilter.tendsto_pure_self
@[simp] theorem Ultrafilter.tendsto_pure_self (b : Ultrafilter α) : Tendsto pure b (𝓝 b) 
Arguments: {α : Type u} (b : Ultrafilter α)
Goal: Filter.Tendsto pure (↑b) (𝓝 b)
-- ultrafilter_comap_pure_nhds
theorem ultrafilter_comap_pure_nhds (b : Ultrafilter α) : comap pure (𝓝 b) ≤ b 
Arguments: {α : Type u} (b : Ultrafilter α)
Goal: Filter.comap pure (𝓝 b) ≤ ↑b
-- ultrafilter_pure_injective
theorem ultrafilter_pure_injective : Function.Injective (pure : α → Ultrafilter α) 
Arguments: {α : Type u}
Goal: Function.Injective pure
-- denseRange_pure
theorem denseRange_pure : DenseRange (pure : α → Ultrafilter α) 
Arguments: {α : Type u}
Goal: DenseRange pure
-- induced_topology_pure
theorem induced_topology_pure :
    TopologicalSpace.induced (pure : α → Ultrafilter α) Ultrafilter.topologicalSpace = ⊥ 
Arguments: {α : Type u}
Goal: TopologicalSpace.induced pure Ultrafilter.topologicalSpace = ⊥
-- denseInducing_pure
theorem denseInducing_pure : @DenseInducing _ _ ⊥ _ (pure : α → Ultrafilter α) 
Arguments: {α : Type u}
Goal: DenseInducing pure
-- denseEmbedding_pure
theorem denseEmbedding_pure : @DenseEmbedding _ _ ⊥ _ (pure : α → Ultrafilter α) 
Arguments: {α : Type u}
Goal: DenseEmbedding pure
-- ultrafilter_extend_extends
theorem ultrafilter_extend_extends (f : α → γ) : Ultrafilter.extend f ∘ pure = f 
Arguments: {α : Type u} {γ : Type u_1} [TopologicalSpace γ] [T2Space γ] (f : α → γ)
Goal: Ultrafilter.extend f ∘ pure = f
-- continuous_ultrafilter_extend
theorem continuous_ultrafilter_extend (f : α → γ) : Continuous (Ultrafilter.extend f) 
Arguments: {α : Type u} {γ : Type u_1} [TopologicalSpace γ] [T2Space γ] [CompactSpace γ] (f : α → γ)
Goal: Continuous (Ultrafilter.extend f)
-- ultrafilter_extend_eq_iff
theorem ultrafilter_extend_eq_iff {f : α → γ} {b : Ultrafilter α} {c : γ} :
    Ultrafilter.extend f b = c ↔ ↑(b.map f) ≤ 𝓝 c 
Arguments: {α : Type u} {γ : Type u_1} [TopologicalSpace γ] [T2Space γ] [CompactSpace γ] {f : α → γ} {b : Ultrafilter α} {c : γ}
Goal: Ultrafilter.extend f b = c ↔ ↑(Ultrafilter.map f b) ≤ 𝓝 c
-- continuous_preStoneCechUnit
theorem continuous_preStoneCechUnit : Continuous (preStoneCechUnit : α → PreStoneCech α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: Continuous preStoneCechUnit
-- denseRange_preStoneCechUnit
theorem denseRange_preStoneCechUnit : DenseRange (preStoneCechUnit : α → PreStoneCech α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: DenseRange preStoneCechUnit
-- preStoneCechCompat
lemma preStoneCechCompat {F G : Ultrafilter α} {x : α} (hF : ↑F ≤ 𝓝 x) (hG : ↑G ≤ 𝓝 x) :
    Ultrafilter.extend g F = Ultrafilter.extend g G 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] [CompactSpace β] {g : α → β} (hg : Continuous g) {F G : Ultrafilter α} {x : α} (hF : ↑F ≤ 𝓝 x) (hG : ↑G ≤ 𝓝 x)
Goal: Ultrafilter.extend g F = Ultrafilter.extend g G
-- preStoneCechExtend_extends
theorem preStoneCechExtend_extends : preStoneCechExtend hg ∘ preStoneCechUnit = g 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] [CompactSpace β] {g : α → β} (hg : Continuous g)
Goal: preStoneCechExtend hg ∘ preStoneCechUnit = g
-- eq_if_preStoneCechUnit_eq
lemma eq_if_preStoneCechUnit_eq {a b : α} (h : preStoneCechUnit a = preStoneCechUnit b) :
    g a = g b 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] [CompactSpace β] {g : α → β} (hg : Continuous g) {a b : α} (h : preStoneCechUnit a = preStoneCechUnit b)
Goal: g a = g b
-- continuous_preStoneCechExtend
theorem continuous_preStoneCechExtend : Continuous (preStoneCechExtend hg) 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] [CompactSpace β] {g : α → β} (hg : Continuous g)
Goal: Continuous (preStoneCechExtend hg)
-- preStoneCech_hom_ext
theorem preStoneCech_hom_ext {g₁ g₂ : PreStoneCech α → β} (h₁ : Continuous g₁) (h₂ : Continuous g₂)
    (h : g₁ ∘ preStoneCechUnit = g₂ ∘ preStoneCechUnit) : g₁ = g₂ 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] {g₁ g₂ : PreStoneCech α → β} (h₁ : Continuous g₁) (h₂ : Continuous g₂) (h : g₁ ∘ preStoneCechUnit = g₂ ∘ preStoneCechUnit)
Goal: g₁ = g₂
-- continuous_stoneCechUnit
theorem continuous_stoneCechUnit : Continuous (stoneCechUnit : α → StoneCech α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: Continuous stoneCechUnit
-- denseRange_stoneCechUnit
theorem denseRange_stoneCechUnit : DenseRange (stoneCechUnit : α → StoneCech α) 
Arguments: {α : Type u} [TopologicalSpace α]
Goal: DenseRange stoneCechUnit
-- stoneCechExtend_extends
theorem stoneCechExtend_extends : stoneCechExtend hg ∘ stoneCechUnit = g 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] [CompactSpace β] {g : α → β} (hg : Continuous g)
Goal: stoneCechExtend hg ∘ stoneCechUnit = g
-- continuous_stoneCechExtend
theorem continuous_stoneCechExtend : Continuous (stoneCechExtend hg) 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] [CompactSpace β] {g : α → β} (hg : Continuous g)
Goal: Continuous (stoneCechExtend hg)
-- eq_if_stoneCechUnit_eq
lemma eq_if_stoneCechUnit_eq {a b : α} {f : α → β} (hcf : Continuous f)
    (h : stoneCechUnit a = stoneCechUnit b) : f a = f b 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] [CompactSpace β] {a b : α} {f : α → β} (hcf : Continuous f) (h : stoneCechUnit a = stoneCechUnit b)
Goal: f a = f b
-- stoneCech_hom_ext
theorem stoneCech_hom_ext {g₁ g₂ : StoneCech α → β} (h₁ : Continuous g₁) (h₂ : Continuous g₂)
    (h : g₁ ∘ stoneCechUnit = g₂ ∘ stoneCechUnit) : g₁ = g₂ 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type v} [TopologicalSpace β] [T2Space β] {g₁ g₂ : StoneCech α → β} (h₁ : Continuous g₁) (h₂ : Continuous g₂) (h : g₁ ∘ stoneCechUnit = g₂ ∘ stoneCechUnit)
Goal: g₁ = g₂
-- PartialDiffeomorph.contMDiffOn_invFun
structure PartialDiffeomorph extends PartialEquiv M N where
  open_source : IsOpen source
  open_target : IsOpen target
  contMDiffOn_toFun : ContMDiffOn I J n toFun source
  contMDiffOn_invFun : ContMDiffOn J I n invFun target
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (self : PartialDiffeomorph I J M N n)
Goal: ContMDiffOn J I n self.invFun self.target
-- PartialDiffeomorph.open_target
structure PartialDiffeomorph extends PartialEquiv M N where
  open_source : IsOpen source
  open_target : IsOpen target
  contMDiffOn_toFun : ContMDiffOn I J n toFun source
  contMDiffOn_invFun : ContMDiffOn J I n invFun target
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (self : PartialDiffeomorph I J M N n)
Goal: IsOpen self.target
-- PartialDiffeomorph.contMDiffOn_toFun
structure PartialDiffeomorph extends PartialEquiv M N where
  open_source : IsOpen source
  open_target : IsOpen target
  contMDiffOn_toFun : ContMDiffOn I J n toFun source
  contMDiffOn_invFun : ContMDiffOn J I n invFun target
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (self : PartialDiffeomorph I J M N n)
Goal: ContMDiffOn I J n (↑self.toPartialEquiv) self.source
-- PartialDiffeomorph.open_source
structure PartialDiffeomorph extends PartialEquiv M N where
  open_source : IsOpen source
  open_target : IsOpen target
  contMDiffOn_toFun : ContMDiffOn I J n toFun source
  contMDiffOn_invFun : ContMDiffOn J I n invFun target
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (self : PartialDiffeomorph I J M N n)
Goal: IsOpen self.source
-- PartialDiffeomorph.contMDiffOn
protected theorem contMDiffOn : ContMDiffOn I J n Φ Φ.source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (Φ : PartialDiffeomorph I J M N n)
Goal: ContMDiffOn I J n (↑Φ.toPartialEquiv) Φ.source
-- PartialDiffeomorph.mdifferentiableOn
protected theorem mdifferentiableOn : MDifferentiableOn I J Φ Φ.source 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (Φ : PartialDiffeomorph I J M N n) (hn : 1 ≤ n)
Goal: MDifferentiableOn I J (↑Φ.toPartialEquiv) Φ.source
-- PartialDiffeomorph.mdifferentiableAt
protected theorem mdifferentiableAt {x : M} (hx : x ∈ Φ.source) : MDifferentiableAt I J Φ x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (Φ : PartialDiffeomorph I J M N n) (hn : 1 ≤ n) {x : M} (hx : x ∈ Φ.source)
Goal: MDifferentiableAt I J (↑Φ.toPartialEquiv) x
-- isLocalDiffeomorphOn_iff
lemma isLocalDiffeomorphOn_iff {f : M → N} (s : Set M) :
    IsLocalDiffeomorphOn I J n f s ↔ ∀ x : s, IsLocalDiffeomorphAt I J n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} (s : Set M)
Goal: IsLocalDiffeomorphOn I J n f s ↔ ∀ (x : ↑s), IsLocalDiffeomorphAt I J n f ↑x
-- isLocalDiffeomorph_iff
lemma isLocalDiffeomorph_iff {f : M → N} :
    IsLocalDiffeomorph I J n f ↔ ∀ x : M, IsLocalDiffeomorphAt I J n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N}
Goal: IsLocalDiffeomorph I J n f ↔ ∀ (x : M), IsLocalDiffeomorphAt I J n f x
-- isLocalDiffeomorph_iff_isLocalDiffeomorphOn_univ
theorem isLocalDiffeomorph_iff_isLocalDiffeomorphOn_univ {f : M → N} :
    IsLocalDiffeomorph I J n f ↔ IsLocalDiffeomorphOn I J n f Set.univ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N}
Goal: IsLocalDiffeomorph I J n f ↔ IsLocalDiffeomorphOn I J n f Set.univ
-- IsLocalDiffeomorph.isLocalDiffeomorphOn
lemma IsLocalDiffeomorph.isLocalDiffeomorphOn
    {f : M → N} (hf : IsLocalDiffeomorph I J n f) (s : Set M) : IsLocalDiffeomorphOn I J n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} (hf : IsLocalDiffeomorph I J n f) (s : Set M)
Goal: IsLocalDiffeomorphOn I J n f s
-- IsLocalDiffeomorphAt.contMDiffAt
lemma IsLocalDiffeomorphAt.contMDiffAt (hf : IsLocalDiffeomorphAt I J n f x) :
    ContMDiffAt I J n f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} {x : M} (hf : IsLocalDiffeomorphAt I J n f x)
Goal: ContMDiffAt I J n f x
-- IsLocalDiffeomorphAt.mdifferentiableAt
lemma IsLocalDiffeomorphAt.mdifferentiableAt (hf : IsLocalDiffeomorphAt I J n f x) (hn : 1 ≤ n) :
    MDifferentiableAt I J f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} {x : M} (hf : IsLocalDiffeomorphAt I J n f x) (hn : 1 ≤ n)
Goal: MDifferentiableAt I J f x
-- IsLocalDiffeomorphOn.contMDiffOn
lemma IsLocalDiffeomorphOn.contMDiffOn (hf : IsLocalDiffeomorphOn I J n f s) :
    ContMDiffOn I J n f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} {s : Set M} (hf : IsLocalDiffeomorphOn I J n f s)
Goal: ContMDiffOn I J n f s
-- IsLocalDiffeomorphOn.mdifferentiableOn
lemma IsLocalDiffeomorphOn.mdifferentiableOn (hf : IsLocalDiffeomorphOn I J n f s) (hn : 1 ≤ n) :
    MDifferentiableOn I J f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} {s : Set M} (hf : IsLocalDiffeomorphOn I J n f s) (hn : 1 ≤ n)
Goal: MDifferentiableOn I J f s
-- IsLocalDiffeomorph.contMDiff
lemma IsLocalDiffeomorph.contMDiff (hf : IsLocalDiffeomorph I J n f) : ContMDiff I J n f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} (hf : IsLocalDiffeomorph I J n f)
Goal: ContMDiff I J n f
-- IsLocalDiffeomorph.mdifferentiable
lemma IsLocalDiffeomorph.mdifferentiable (hf : IsLocalDiffeomorph I J n f) (hn : 1 ≤ n) :
    MDifferentiable I J f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} (hf : IsLocalDiffeomorph I J n f) (hn : 1 ≤ n)
Goal: MDifferentiable I J f
-- Diffeomorph.isLocalDiffeomorph
lemma Diffeomorph.isLocalDiffeomorph (Φ : M ≃ₘ^n⟮I, J⟯ N) : IsLocalDiffeomorph I J n Φ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (Φ : M ≃ₘ^n⟮I, J⟯ N)
Goal: IsLocalDiffeomorph I J n ⇑Φ
-- IsLocalDiffeomorphOn.isLocalHomeomorphOn
theorem IsLocalDiffeomorphOn.isLocalHomeomorphOn {s : Set M} (hf : IsLocalDiffeomorphOn I J n f s) :
    IsLocalHomeomorphOn f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} {s : Set M} (hf : IsLocalDiffeomorphOn I J n f s)
Goal: IsLocalHomeomorphOn f s
-- IsLocalDiffeomorph.isLocalHomeomorph
theorem IsLocalDiffeomorph.isLocalHomeomorph (hf : IsLocalDiffeomorph I J n f) :
    IsLocalHomeomorph f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} (hf : IsLocalDiffeomorph I J n f)
Goal: IsLocalHomeomorph f
-- IsLocalDiffeomorph.isOpenMap
lemma IsLocalDiffeomorph.isOpenMap (hf : IsLocalDiffeomorph I J n f) : IsOpenMap f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} (hf : IsLocalDiffeomorph I J n f)
Goal: IsOpenMap f
-- IsLocalDiffeomorph.isOpen_range
lemma IsLocalDiffeomorph.isOpen_range (hf : IsLocalDiffeomorph I J n f) : IsOpen (range f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} (hf : IsLocalDiffeomorph I J n f)
Goal: IsOpen (Set.range f)
-- IsLocalDiffeomorph.image_coe
lemma IsLocalDiffeomorph.image_coe (hf : IsLocalDiffeomorph I J n f) : hf.image.1 = range f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_4} [TopologicalSpace H] {G : Type u_5} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_6} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_7} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} {f : M → N} (hf : IsLocalDiffeomorph I J n f)
Goal: hf.image.carrier = Set.range f
-- ModelWithCorners.isInteriorPoint_iff
lemma isInteriorPoint_iff {x : M} :
    I.IsInteriorPoint x ↔ extChartAt I x x ∈ interior (extChartAt I x).target 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {x : M}
Goal: I.IsInteriorPoint x ↔ ↑(extChartAt I x) x ∈ interior (extChartAt I x).target
-- ModelWithCorners.isBoundaryPoint_iff
lemma isBoundaryPoint_iff {x : M} : I.IsBoundaryPoint x ↔ extChartAt I x x ∈ frontier (range I) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {x : M}
Goal: I.IsBoundaryPoint x ↔ ↑(extChartAt I x) x ∈ frontier (Set.range ↑I)
-- ModelWithCorners.isInteriorPoint_or_isBoundaryPoint
lemma isInteriorPoint_or_isBoundaryPoint (x : M) : I.IsInteriorPoint x ∨ I.IsBoundaryPoint x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] (x : M)
Goal: I.IsInteriorPoint x ∨ I.IsBoundaryPoint x
-- ModelWithCorners.interior_union_boundary_eq_univ
lemma interior_union_boundary_eq_univ : (I.interior M) ∪ (I.boundary M) = (univ : Set M) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M]
Goal: I.interior M ∪ I.boundary M = Set.univ