ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- exists_isCompact_isClosed_nhds_zero
@[to_additive (attr := deprecated exists_mem_nhds_isCompact_isClosed (since := "2024-01-28"))]
theorem exists_isCompact_isClosed_nhds_one [WeaklyLocallyCompactSpace G] :
    ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 
Arguments: (G : Type w) [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] [WeaklyLocallyCompactSpace G]
Goal: ∃ K, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 0
-- exists_isCompact_isClosed_nhds_one
@[to_additive (attr := deprecated exists_mem_nhds_isCompact_isClosed (since := "2024-01-28"))]
theorem exists_isCompact_isClosed_nhds_one [WeaklyLocallyCompactSpace G] :
    ∃ K : Set G, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1 
Arguments: (G : Type w) [TopologicalSpace G] [Group G] [TopologicalGroup G] [WeaklyLocallyCompactSpace G]
Goal: ∃ K, IsCompact K ∧ IsClosed K ∧ K ∈ 𝓝 1
-- nhds_add
@[to_additive]
theorem nhds_mul (x y : G) : 𝓝 (x * y) = 𝓝 x * 𝓝 y 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (x y : G)
Goal: 𝓝 (x + y) = 𝓝 x + 𝓝 y
-- nhds_mul
@[to_additive]
theorem nhds_mul (x y : G) : 𝓝 (x * y) = 𝓝 x * 𝓝 y 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (x y : G)
Goal: 𝓝 (x * y) = 𝓝 x * 𝓝 y
-- nhdsAddHom_apply
@[to_additive (attr := simps)
  "On an additive topological group, `𝓝 : G → Filter G` can be promoted to an `AddHom`."]
def nhdsMulHom : G →ₙ* Filter G 
Arguments: {G : Type w} [TopologicalSpace G] [AddGroup G] [TopologicalAddGroup G] (x : G)
Goal: nhdsAddHom x = 𝓝 x
-- nhdsMulHom_apply
@[to_additive (attr := simps)
  "On an additive topological group, `𝓝 : G → Filter G` can be promoted to an `AddHom`."]
def nhdsMulHom : G →ₙ* Filter G 
Arguments: {G : Type w} [TopologicalSpace G] [Group G] [TopologicalGroup G] (x : G)
Goal: nhdsMulHom x = 𝓝 x
-- QuotientAddGroup.continuous_smul₁
@[to_additive]
theorem QuotientGroup.continuous_smul₁ (x : G ⧸ Γ) : Continuous fun g : G => g • x 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [ContinuousAdd G] {Γ : AddSubgroup G} (x : G ⧸ Γ)
Goal: Continuous fun g => g +ᵥ x
-- QuotientGroup.continuous_smul₁
@[to_additive]
theorem QuotientGroup.continuous_smul₁ (x : G ⧸ Γ) : Continuous fun g : G => g • x 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [ContinuousMul G] {Γ : Subgroup G} (x : G ⧸ Γ)
Goal: Continuous fun g => g • x
-- Units.embedding_val
@[to_additive] theorem Units.embedding_val [Group G] [TopologicalSpace G] [ContinuousInv G] :
    Embedding (val : Gˣ → G) 
Arguments: {G : Type w} [Group G] [TopologicalSpace G] [ContinuousInv G]
Goal: Embedding Units.val
-- AddUnits.embedding_val
@[to_additive] theorem Units.embedding_val [Group G] [TopologicalSpace G] [ContinuousInv G] :
    Embedding (val : Gˣ → G) 
Arguments: {G : Type w} [AddGroup G] [TopologicalSpace G] [ContinuousNeg G]
Goal: Embedding AddUnits.val
-- topologicalGroup_sInf
@[to_additive]
theorem topologicalGroup_sInf {ts : Set (TopologicalSpace G)}
    (h : ∀ t ∈ ts, @TopologicalGroup G t _) : @TopologicalGroup G (sInf ts) _ 
Arguments: {G : Type w} [Group G] {ts : Set (TopologicalSpace G)} (h : ∀ t ∈ ts, TopologicalGroup G)
Goal: TopologicalGroup G
-- topologicalAddGroup_sInf
@[to_additive]
theorem topologicalGroup_sInf {ts : Set (TopologicalSpace G)}
    (h : ∀ t ∈ ts, @TopologicalGroup G t _) : @TopologicalGroup G (sInf ts) _ 
Arguments: {G : Type w} [AddGroup G] {ts : Set (TopologicalSpace G)} (h : ∀ t ∈ ts, TopologicalAddGroup G)
Goal: TopologicalAddGroup G
-- topologicalGroup_iInf
@[to_additive]
theorem topologicalGroup_iInf {ts' : ι → TopologicalSpace G}
    (h' : ∀ i, @TopologicalGroup G (ts' i) _) : @TopologicalGroup G (⨅ i, ts' i) _ 
Arguments: {G : Type w} {ι : Sort u_1} [Group G] {ts' : ι → TopologicalSpace G} (h' : ∀ (i : ι), TopologicalGroup G)
Goal: TopologicalGroup G
-- topologicalAddGroup_iInf
@[to_additive]
theorem topologicalGroup_iInf {ts' : ι → TopologicalSpace G}
    (h' : ∀ i, @TopologicalGroup G (ts' i) _) : @TopologicalGroup G (⨅ i, ts' i) _ 
Arguments: {G : Type w} {ι : Sort u_1} [AddGroup G] {ts' : ι → TopologicalSpace G} (h' : ∀ (i : ι), TopologicalAddGroup G)
Goal: TopologicalAddGroup G
-- topologicalAddGroup_inf
@[to_additive]
theorem topologicalGroup_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @TopologicalGroup G t₁ _)
    (h₂ : @TopologicalGroup G t₂ _) : @TopologicalGroup G (t₁ ⊓ t₂) _ 
Arguments: {G : Type w} [AddGroup G] {t₁ t₂ : TopologicalSpace G} (h₁ : TopologicalAddGroup G) (h₂ : TopologicalAddGroup G)
Goal: TopologicalAddGroup G
-- topologicalGroup_inf
@[to_additive]
theorem topologicalGroup_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @TopologicalGroup G t₁ _)
    (h₂ : @TopologicalGroup G t₂ _) : @TopologicalGroup G (t₁ ⊓ t₂) _ 
Arguments: {G : Type w} [Group G] {t₁ t₂ : TopologicalSpace G} (h₁ : TopologicalGroup G) (h₂ : TopologicalGroup G)
Goal: TopologicalGroup G
-- GroupTopology.continuous_mul'
@[to_additive "A version of the global `continuous_add` suitable for dot notation."]
theorem continuous_mul' (g : GroupTopology α) :
    haveI := g.toTopologicalSpace
    Continuous fun p : α × α => p.1 * p.2 
Arguments: {α : Type u} [Group α] (g : GroupTopology α)
Goal: Continuous fun p => p.1 * p.2
-- AddGroupTopology.continuous_add'
@[to_additive "A version of the global `continuous_add` suitable for dot notation."]
theorem continuous_mul' (g : GroupTopology α) :
    haveI := g.toTopologicalSpace
    Continuous fun p : α × α => p.1 * p.2 
Arguments: {α : Type u} [AddGroup α] (g : AddGroupTopology α)
Goal: Continuous fun p => p.1 + p.2
-- AddGroupTopology.continuous_neg'
@[to_additive "A version of the global `continuous_neg` suitable for dot notation."]
theorem continuous_inv' (g : GroupTopology α) :
    haveI := g.toTopologicalSpace
    Continuous (Inv.inv : α → α) 
Arguments: {α : Type u} [AddGroup α] (g : AddGroupTopology α)
Goal: Continuous Neg.neg
-- GroupTopology.continuous_inv'
@[to_additive "A version of the global `continuous_neg` suitable for dot notation."]
theorem continuous_inv' (g : GroupTopology α) :
    haveI := g.toTopologicalSpace
    Continuous (Inv.inv : α → α) 
Arguments: {α : Type u} [Group α] (g : GroupTopology α)
Goal: Continuous Inv.inv
-- GroupTopology.toTopologicalSpace_injective
@[to_additive]
theorem toTopologicalSpace_injective :
    Function.Injective (toTopologicalSpace : GroupTopology α → TopologicalSpace α) 
Arguments: {α : Type u} [Group α]
Goal: Function.Injective GroupTopology.toTopologicalSpace
-- AddGroupTopology.toTopologicalSpace_injective
@[to_additive]
theorem toTopologicalSpace_injective :
    Function.Injective (toTopologicalSpace : GroupTopology α → TopologicalSpace α) 
Arguments: {α : Type u} [AddGroup α]
Goal: Function.Injective AddGroupTopology.toTopologicalSpace
-- GroupTopology.ext'
@[to_additive (attr := ext)]
theorem ext' {f g : GroupTopology α} (h : f.IsOpen = g.IsOpen) : f = g 
Arguments: {α : Type u} [Group α] {f g : GroupTopology α} (h : TopologicalSpace.IsOpen = TopologicalSpace.IsOpen)
Goal: f = g
-- AddGroupTopology.ext'
@[to_additive (attr := ext)]
theorem ext' {f g : GroupTopology α} (h : f.IsOpen = g.IsOpen) : f = g 
Arguments: {α : Type u} [AddGroup α] {f g : AddGroupTopology α} (h : TopologicalSpace.IsOpen = TopologicalSpace.IsOpen)
Goal: f = g
-- GroupTopology.toTopologicalSpace_le
@[to_additive (attr := simp)]
theorem toTopologicalSpace_le {x y : GroupTopology α} :
    x.toTopologicalSpace ≤ y.toTopologicalSpace ↔ x ≤ y 
Arguments: {α : Type u} [Group α] {x y : GroupTopology α}
Goal: x.toTopologicalSpace ≤ y.toTopologicalSpace ↔ x ≤ y
-- AddGroupTopology.toTopologicalSpace_le
@[to_additive (attr := simp)]
theorem toTopologicalSpace_le {x y : GroupTopology α} :
    x.toTopologicalSpace ≤ y.toTopologicalSpace ↔ x ≤ y 
Arguments: {α : Type u} [AddGroup α] {x y : AddGroupTopology α}
Goal: x.toTopologicalSpace ≤ y.toTopologicalSpace ↔ x ≤ y
-- GroupTopology.toTopologicalSpace_top
@[to_additive (attr := simp)]
theorem toTopologicalSpace_top : (⊤ : GroupTopology α).toTopologicalSpace = ⊤ 
Arguments: {α : Type u} [Group α]
Goal: ⊤.toTopologicalSpace = ⊤
-- AddGroupTopology.toTopologicalSpace_top
@[to_additive (attr := simp)]
theorem toTopologicalSpace_top : (⊤ : GroupTopology α).toTopologicalSpace = ⊤ 
Arguments: {α : Type u} [AddGroup α]
Goal: ⊤.toTopologicalSpace = ⊤
-- GroupTopology.toTopologicalSpace_bot
@[to_additive (attr := simp)]
theorem toTopologicalSpace_bot : (⊥ : GroupTopology α).toTopologicalSpace = ⊥ 
Arguments: {α : Type u} [Group α]
Goal: ⊥.toTopologicalSpace = ⊥
-- AddGroupTopology.toTopologicalSpace_bot
@[to_additive (attr := simp)]
theorem toTopologicalSpace_bot : (⊥ : GroupTopology α).toTopologicalSpace = ⊥ 
Arguments: {α : Type u} [AddGroup α]
Goal: ⊥.toTopologicalSpace = ⊥
-- AddGroupTopology.toTopologicalSpace_inf
@[to_additive (attr := simp)]
theorem toTopologicalSpace_inf (x y : GroupTopology α) :
    (x ⊓ y).toTopologicalSpace = x.toTopologicalSpace ⊓ y.toTopologicalSpace 
Arguments: {α : Type u} [AddGroup α] (x y : AddGroupTopology α)
Goal: (x ⊓ y).toTopologicalSpace = x.toTopologicalSpace ⊓ y.toTopologicalSpace
-- GroupTopology.toTopologicalSpace_inf
@[to_additive (attr := simp)]
theorem toTopologicalSpace_inf (x y : GroupTopology α) :
    (x ⊓ y).toTopologicalSpace = x.toTopologicalSpace ⊓ y.toTopologicalSpace 
Arguments: {α : Type u} [Group α] (x y : GroupTopology α)
Goal: (x ⊓ y).toTopologicalSpace = x.toTopologicalSpace ⊓ y.toTopologicalSpace
-- GroupTopology.toTopologicalSpace_sInf
@[to_additive (attr := simp)]
theorem toTopologicalSpace_sInf (s : Set (GroupTopology α)) :
    (sInf s).toTopologicalSpace = sInf (toTopologicalSpace '' s) 
Arguments: {α : Type u} [Group α] (s : Set (GroupTopology α))
Goal: (sInf s).toTopologicalSpace = sInf (GroupTopology.toTopologicalSpace '' s)
-- AddGroupTopology.toTopologicalSpace_sInf
@[to_additive (attr := simp)]
theorem toTopologicalSpace_sInf (s : Set (GroupTopology α)) :
    (sInf s).toTopologicalSpace = sInf (toTopologicalSpace '' s) 
Arguments: {α : Type u} [AddGroup α] (s : Set (AddGroupTopology α))
Goal: (sInf s).toTopologicalSpace = sInf (AddGroupTopology.toTopologicalSpace '' s)
-- AddGroupTopology.toTopologicalSpace_iInf
@[to_additive (attr := simp)]
theorem toTopologicalSpace_iInf {ι} (s : ι → GroupTopology α) :
    (⨅ i, s i).toTopologicalSpace = ⨅ i, (s i).toTopologicalSpace 
Arguments: {α : Type u} [AddGroup α] {ι : Sort u_1} (s : ι → AddGroupTopology α)
Goal: (⨅ i, s i).toTopologicalSpace = ⨅ i, (s i).toTopologicalSpace
-- GroupTopology.toTopologicalSpace_iInf
@[to_additive (attr := simp)]
theorem toTopologicalSpace_iInf {ι} (s : ι → GroupTopology α) :
    (⨅ i, s i).toTopologicalSpace = ⨅ i, (s i).toTopologicalSpace 
Arguments: {α : Type u} [Group α] {ι : Sort u_1} (s : ι → GroupTopology α)
Goal: (⨅ i, s i).toTopologicalSpace = ⨅ i, (s i).toTopologicalSpace
-- GroupTopology.coinduced_continuous
@[to_additive]
theorem coinduced_continuous {α β : Type*} [t : TopologicalSpace α] [Group β] (f : α → β) :
    Continuous[t, (coinduced f).toTopologicalSpace] f 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace α] [Group β] (f : α → β)
Goal: Continuous f
-- AddGroupTopology.coinduced_continuous
@[to_additive]
theorem coinduced_continuous {α β : Type*} [t : TopologicalSpace α] [Group β] (f : α → β) :
    Continuous[t, (coinduced f).toTopologicalSpace] f 
Arguments: {α : Type u_1} {β : Type u_2} [t : TopologicalSpace α] [AddGroup β] (f : α → β)
Goal: Continuous f
-- Diffeomorph.contMDiff_invFun
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure Diffeomorph extends M ≃ M' where
  protected contMDiff_toFun : ContMDiff I I' n toEquiv
  protected contMDiff_invFun : ContMDiff I' I n toEquiv.symm
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (self : M ≃ₘ^n⟮I, I'⟯ M')
Goal: ContMDiff I' I n ⇑self.symm
-- Diffeomorph.contMDiff_toFun
-- Porting note(#5171): was @[nolint has_nonempty_instance]
structure Diffeomorph extends M ≃ M' where
  protected contMDiff_toFun : ContMDiff I I' n toEquiv
  protected contMDiff_invFun : ContMDiff I' I n toEquiv.symm
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (self : M ≃ₘ^n⟮I, I'⟯ M')
Goal: ContMDiff I I' n ⇑self.toEquiv
-- Diffeomorph.toEquiv_injective
theorem toEquiv_injective : Injective (Diffeomorph.toEquiv : (M ≃ₘ^n⟮I, I'⟯ M') → M ≃ M')
  
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞}
Goal: Function.Injective Diffeomorph.toEquiv
-- Diffeomorph.continuous
@[continuity]
protected theorem continuous (h : M ≃ₘ^n⟮I, I'⟯ M') : Continuous h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M')
Goal: Continuous ⇑h
-- Diffeomorph.contMDiff
protected theorem contMDiff (h : M ≃ₘ^n⟮I, I'⟯ M') : ContMDiff I I' n h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M')
Goal: ContMDiff I I' n ⇑h
-- Diffeomorph.contMDiffAt
protected theorem contMDiffAt (h : M ≃ₘ^n⟮I, I'⟯ M') {x} : ContMDiffAt I I' n h x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M') {x : M}
Goal: ContMDiffAt I I' n (⇑h) x
-- Diffeomorph.contMDiffWithinAt
protected theorem contMDiffWithinAt (h : M ≃ₘ^n⟮I, I'⟯ M') {s x} : ContMDiffWithinAt I I' n h s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M') {s : Set M} {x : M}
Goal: ContMDiffWithinAt I I' n (⇑h) s x
-- Diffeomorph.contDiff
protected theorem contDiff (h : E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E') : ContDiff 𝕜 n h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {n : ℕ∞} (h : E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E')
Goal: ContDiff 𝕜 n ⇑h
-- Diffeomorph.smooth
protected theorem smooth (h : M ≃ₘ⟮I, I'⟯ M') : Smooth I I' h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] (h : M ≃ₘ⟮I, I'⟯ M')
Goal: Smooth I I' ⇑h
-- Diffeomorph.mdifferentiable
protected theorem mdifferentiable (h : M ≃ₘ^n⟮I, I'⟯ M') (hn : 1 ≤ n) : MDifferentiable I I' h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M') (hn : 1 ≤ n)
Goal: MDifferentiable I I' ⇑h
-- Diffeomorph.mdifferentiableOn
protected theorem mdifferentiableOn (h : M ≃ₘ^n⟮I, I'⟯ M') (s : Set M) (hn : 1 ≤ n) :
    MDifferentiableOn I I' h s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M') (s : Set M) (hn : 1 ≤ n)
Goal: MDifferentiableOn I I' (⇑h) s
-- Diffeomorph.coe_toEquiv
@[simp]
theorem coe_toEquiv (h : M ≃ₘ^n⟮I, I'⟯ M') : ⇑h.toEquiv = h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M')
Goal: ⇑h.toEquiv = ⇑h
-- Diffeomorph.coe_coe
@[simp, norm_cast]
theorem coe_coe (h : M ≃ₘ^n⟮I, I'⟯ M') : ⇑(h : C^n⟮I, M; I', M'⟯) = h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M')
Goal: ⇑↑h = ⇑h
-- Diffeomorph.toEquiv_inj
@[simp]
theorem toEquiv_inj {h h' : M ≃ₘ^n⟮I, I'⟯ M'} : h.toEquiv = h'.toEquiv ↔ h = h' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} {h h' : M ≃ₘ^n⟮I, I'⟯ M'}
Goal: h.toEquiv = h'.toEquiv ↔ h = h'
-- Diffeomorph.coeFn_injective
theorem coeFn_injective : Injective ((↑) : (M ≃ₘ^n⟮I, I'⟯ M') → (M → M')) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞}
Goal: Function.Injective DFunLike.coe
-- Diffeomorph.ext
@[ext]
theorem ext {h h' : M ≃ₘ^n⟮I, I'⟯ M'} (Heq : ∀ x, h x = h' x) : h = h' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} {h h' : M ≃ₘ^n⟮I, I'⟯ M'} (Heq : ∀ (x : M), h x = h' x)
Goal: h = h'
-- Diffeomorph.refl_toEquiv
@[simp]
theorem refl_toEquiv : (Diffeomorph.refl I M n).toEquiv = Equiv.refl _ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_9) [TopologicalSpace M] [ChartedSpace H M] (n : ℕ∞)
Goal: (Diffeomorph.refl I M n).toEquiv = Equiv.refl M
-- Diffeomorph.coe_refl
@[simp]
theorem coe_refl : ⇑(Diffeomorph.refl I M n) = id 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_5} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (M : Type u_9) [TopologicalSpace M] [ChartedSpace H M] (n : ℕ∞)
Goal: ⇑(Diffeomorph.refl I M n) = id
-- Diffeomorph.trans_refl
@[simp]
theorem trans_refl (h : M ≃ₘ^n⟮I, I'⟯ M') : h.trans (Diffeomorph.refl I' M' n) = h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M')
Goal: h.trans (Diffeomorph.refl I' M' n) = h
-- Diffeomorph.refl_trans
@[simp]
theorem refl_trans (h : M ≃ₘ^n⟮I, I'⟯ M') : (Diffeomorph.refl I M n).trans h = h 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {n : ℕ∞} (h : M ≃ₘ^n⟮I, I'⟯ M')
Goal: (Diffeomorph.refl I M n).trans h = h
-- Diffeomorph.coe_trans
@[simp]
theorem coe_trans (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : M' ≃ₘ^n⟮I', J⟯ N) : ⇑(h₁.trans h₂) = h₂ ∘ h₁ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {E' : Type u_3} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {H' : Type u_6} [TopologicalSpace H'] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {M' : Type u_10} [TopologicalSpace M'] [ChartedSpace H' M'] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : M' ≃ₘ^n⟮I', J⟯ N)
Goal: ⇑(h₁.trans h₂) = ⇑h₂ ∘ ⇑h₁
-- Diffeomorph.apply_symm_apply
@[simp]
theorem apply_symm_apply (h : M ≃ₘ^n⟮I, J⟯ N) (x : N) : h (h.symm x) = x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type u_5} [TopologicalSpace H] {G : Type u_7} [TopologicalSpace G] {I : ModelWithCorners 𝕜 E H} {J : ModelWithCorners 𝕜 F G} {M : Type u_9} [TopologicalSpace M] [ChartedSpace H M] {N : Type u_11} [TopologicalSpace N] [ChartedSpace G N] {n : ℕ∞} (h : M ≃ₘ^n⟮I, J⟯ N) (x : N)
Goal: h (h.symm x) = x