ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- EuclideanQuadrant.ext
@[ext]
theorem EuclideanQuadrant.ext (x y : EuclideanQuadrant n) (h : x.1 = y.1) : x = y 
Arguments: {n : ℕ} (x y : EuclideanQuadrant n) (h : ↑x = ↑y)
Goal: x = y
-- EuclideanHalfSpace.ext
@[ext]
theorem EuclideanHalfSpace.ext [Zero (Fin n)] (x y : EuclideanHalfSpace n)
    (h : x.1 = y.1) : x = y 
Arguments: {n : ℕ} [Zero (Fin n)] (x y : EuclideanHalfSpace n) (h : ↑x = ↑y)
Goal: x = y
-- range_euclideanHalfSpace
theorem range_euclideanHalfSpace (n : ℕ) [Zero (Fin n)] :
    (range fun x : EuclideanHalfSpace n => x.val) = { y | 0 ≤ y 0 } 
Arguments: (n : ℕ) [Zero (Fin n)]
Goal: (Set.range fun x => ↑x) = {y | 0 ≤ y 0}
-- range_euclideanQuadrant
theorem range_euclideanQuadrant (n : ℕ) :
    (range fun x : EuclideanQuadrant n => x.val) = { y | ∀ i : Fin n, 0 ≤ y i } 
Arguments: (n : ℕ)
Goal: (Set.range fun x => ↑x) = {y | ∀ (i : Fin n), 0 ≤ y i}
-- SmoothBumpCovering.embeddingPiTangent_coe
@[local simp]
theorem embeddingPiTangent_coe :
    ⇑f.embeddingPiTangent = fun x i => (f i x • extChartAt I (f.c i) x, f i x) 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [T2Space M] [hi : Fintype ι] {s : Set M} (f : SmoothBumpCovering ι I M s)
Goal: ⇑f.embeddingPiTangent = fun x i => (↑(f.toFun i) x • ↑(extChartAt I (f.c i)) x, ↑(f.toFun i) x)
-- SmoothBumpCovering.embeddingPiTangent_injOn
theorem embeddingPiTangent_injOn : InjOn f.embeddingPiTangent s 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [T2Space M] [hi : Fintype ι] {s : Set M} (f : SmoothBumpCovering ι I M s)
Goal: Set.InjOn (⇑f.embeddingPiTangent) s
-- SmoothBumpCovering.embeddingPiTangent_injective
theorem embeddingPiTangent_injective (f : SmoothBumpCovering ι I M) :
    Injective f.embeddingPiTangent 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [T2Space M] [hi : Fintype ι] (f : SmoothBumpCovering ι I M)
Goal: Function.Injective ⇑f.embeddingPiTangent
-- SmoothBumpCovering.comp_embeddingPiTangent_mfderiv
theorem comp_embeddingPiTangent_mfderiv (x : M) (hx : x ∈ s) :
    ((ContinuousLinearMap.fst ℝ E ℝ).comp
            (@ContinuousLinearMap.proj ℝ _ ι (fun _ => E × ℝ) _ _ (fun _ => inferInstance)
              (f.ind x hx))).comp
        (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) =
      mfderiv I I (chartAt H (f.c (f.ind x hx))) x 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [T2Space M] [hi : Fintype ι] {s : Set M} (f : SmoothBumpCovering ι I M s) (x : M) (hx : x ∈ s)
Goal: ((ContinuousLinearMap.fst ℝ E ℝ).comp (ContinuousLinearMap.proj (f.ind x hx))).comp
    (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x) =
  mfderiv I I (↑(chartAt H (f.c (f.ind x hx)))) x
-- SmoothBumpCovering.embeddingPiTangent_ker_mfderiv
theorem embeddingPiTangent_ker_mfderiv (x : M) (hx : x ∈ s) :
    LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) = ⊥ 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [T2Space M] [hi : Fintype ι] {s : Set M} (f : SmoothBumpCovering ι I M s) (x : M) (hx : x ∈ s)
Goal: LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x) = ⊥
-- SmoothBumpCovering.embeddingPiTangent_injective_mfderiv
theorem embeddingPiTangent_injective_mfderiv (x : M) (hx : x ∈ s) :
    Injective (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [T2Space M] [hi : Fintype ι] {s : Set M} (f : SmoothBumpCovering ι I M s) (x : M) (hx : x ∈ s)
Goal: Function.Injective ⇑(mfderiv I 𝓘(ℝ, ι → E × ℝ) (⇑f.embeddingPiTangent) x)
-- SmoothBumpCovering.exists_immersion_euclidean
theorem exists_immersion_euclidean [Finite ι] (f : SmoothBumpCovering ι I M) :
    ∃ (n : ℕ) (e : M → EuclideanSpace ℝ (Fin n)),
      Smooth I (𝓡 n) e ∧ Injective e ∧ ∀ x : M, Injective (mfderiv I (𝓡 n) e x) 
Arguments: {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [T2Space M] [Finite ι] (f : SmoothBumpCovering ι I M)
Goal: ∃ n e,
  Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧
    Function.Injective e ∧ ∀ (x : M), Function.Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)
-- exists_embedding_euclidean_of_compact
theorem exists_embedding_euclidean_of_compact [T2Space M] [CompactSpace M] :
    ∃ (n : ℕ) (e : M → EuclideanSpace ℝ (Fin n)),
      Smooth I (𝓡 n) e ∧ ClosedEmbedding e ∧ ∀ x : M, Injective (mfderiv I (𝓡 n) e x) 
Arguments: {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M] [T2Space M] [CompactSpace M]
Goal: ∃ n e,
  Smooth I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e ∧
    ClosedEmbedding e ∧ ∀ (x : M), Function.Injective ⇑(mfderiv I 𝓘(ℝ, EuclideanSpace ℝ (Fin n)) e x)
-- PartialHomeomorph.continuousOn_toFun
-- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance]
structure PartialHomeomorph (X : Type*) (Y : Type*) [TopologicalSpace X]
  [TopologicalSpace Y] extends PartialEquiv X Y where
  open_source : IsOpen source
  open_target : IsOpen target
  continuousOn_toFun : ContinuousOn toFun source
  continuousOn_invFun : ContinuousOn invFun target
Arguments: {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [TopologicalSpace Y] (self : PartialHomeomorph X Y)
Goal: ContinuousOn (↑self.toPartialEquiv) self.source
-- PartialHomeomorph.continuousOn_invFun
-- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance]
structure PartialHomeomorph (X : Type*) (Y : Type*) [TopologicalSpace X]
  [TopologicalSpace Y] extends PartialEquiv X Y where
  open_source : IsOpen source
  open_target : IsOpen target
  continuousOn_toFun : ContinuousOn toFun source
  continuousOn_invFun : ContinuousOn invFun target
Arguments: {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [TopologicalSpace Y] (self : PartialHomeomorph X Y)
Goal: ContinuousOn self.invFun self.target
-- PartialHomeomorph.open_source
-- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance]
structure PartialHomeomorph (X : Type*) (Y : Type*) [TopologicalSpace X]
  [TopologicalSpace Y] extends PartialEquiv X Y where
  open_source : IsOpen source
  open_target : IsOpen target
  continuousOn_toFun : ContinuousOn toFun source
  continuousOn_invFun : ContinuousOn invFun target
Arguments: {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [TopologicalSpace Y] (self : PartialHomeomorph X Y)
Goal: IsOpen self.source
-- PartialHomeomorph.open_target
-- Porting note(#5171): this linter isn't ported yet. @[nolint has_nonempty_instance]
structure PartialHomeomorph (X : Type*) (Y : Type*) [TopologicalSpace X]
  [TopologicalSpace Y] extends PartialEquiv X Y where
  open_source : IsOpen source
  open_target : IsOpen target
  continuousOn_toFun : ContinuousOn toFun source
  continuousOn_invFun : ContinuousOn invFun target
Arguments: {X : Type u_7} {Y : Type u_8} [TopologicalSpace X] [TopologicalSpace Y] (self : PartialHomeomorph X Y)
Goal: IsOpen self.target
-- PartialHomeomorph.continuousOn
protected theorem continuousOn : ContinuousOn e e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ContinuousOn (↑e) e.source
-- PartialHomeomorph.continuousOn_symm
theorem continuousOn_symm : ContinuousOn e.symm e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ContinuousOn (↑e.symm) e.target
-- PartialHomeomorph.mk_coe
@[simp, mfld_simps]
theorem mk_coe (e : PartialEquiv X Y) (a b c d) : (PartialHomeomorph.mk e a b c d : X → Y) = e 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialEquiv X Y) (a : IsOpen e.source) (b : IsOpen e.target) (c : ContinuousOn (↑e) e.source) (d : ContinuousOn e.invFun e.target)
Goal: ↑{ toPartialEquiv := e, open_source := a, open_target := b, continuousOn_toFun := c, continuousOn_invFun := d } = ↑e
-- PartialHomeomorph.mk_coe_symm
@[simp, mfld_simps]
theorem mk_coe_symm (e : PartialEquiv X Y) (a b c d) :
    ((PartialHomeomorph.mk e a b c d).symm : Y → X) = e.symm 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialEquiv X Y) (a : IsOpen e.source) (b : IsOpen e.target) (c : ContinuousOn (↑e) e.source) (d : ContinuousOn e.invFun e.target)
Goal: ↑{ toPartialEquiv := e, open_source := a, open_target := b, continuousOn_toFun := c, continuousOn_invFun := d }.symm =
  ↑e.symm
-- PartialHomeomorph.toPartialEquiv_injective
theorem toPartialEquiv_injective :
    Injective (toPartialEquiv : PartialHomeomorph X Y → PartialEquiv X Y)
  
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Function.Injective PartialHomeomorph.toPartialEquiv
-- PartialHomeomorph.toFun_eq_coe
@[simp, mfld_simps]
theorem toFun_eq_coe (e : PartialHomeomorph X Y) : e.toFun = e 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ↑e.toPartialEquiv = ↑e
-- PartialHomeomorph.invFun_eq_coe
@[simp, mfld_simps]
theorem invFun_eq_coe (e : PartialHomeomorph X Y) : e.invFun = e.symm 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: e.invFun = ↑e.symm
-- PartialHomeomorph.coe_coe
@[simp, mfld_simps]
theorem coe_coe : (e.toPartialEquiv : X → Y) = e 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ↑e.toPartialEquiv = ↑e
-- PartialHomeomorph.coe_coe_symm
@[simp, mfld_simps]
theorem coe_coe_symm : (e.toPartialEquiv.symm : Y → X) = e.symm 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ↑e.symm = ↑e.symm
-- PartialHomeomorph.map_source
@[simp, mfld_simps]
theorem map_source {x : X} (h : x ∈ e.source) : e x ∈ e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (h : x ∈ e.source)
Goal: ↑e x ∈ e.target
-- PartialHomeomorph.map_source''
lemma map_source'' : e '' e.source ⊆ e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: ↑e '' e.source ⊆ e.target
-- PartialHomeomorph.map_target
@[simp, mfld_simps]
theorem map_target {x : Y} (h : x ∈ e.target) : e.symm x ∈ e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : Y} (h : x ∈ e.target)
Goal: ↑e.symm x ∈ e.source
-- PartialHomeomorph.left_inv
@[simp, mfld_simps]
theorem left_inv {x : X} (h : x ∈ e.source) : e.symm (e x) = x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (h : x ∈ e.source)
Goal: ↑e.symm (↑e x) = x
-- PartialHomeomorph.right_inv
@[simp, mfld_simps]
theorem right_inv {x : Y} (h : x ∈ e.target) : e (e.symm x) = x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : Y} (h : x ∈ e.target)
Goal: ↑e (↑e.symm x) = x
-- PartialHomeomorph.eq_symm_apply
theorem eq_symm_apply {x : X} {y : Y} (hx : x ∈ e.source) (hy : y ∈ e.target) :
    x = e.symm y ↔ e x = y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} {y : Y} (hx : x ∈ e.source) (hy : y ∈ e.target)
Goal: x = ↑e.symm y ↔ ↑e x = y
-- PartialHomeomorph.mapsTo
protected theorem mapsTo : MapsTo e e.source e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: Set.MapsTo (↑e) e.source e.target
-- PartialHomeomorph.symm_mapsTo
protected theorem symm_mapsTo : MapsTo e.symm e.target e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: Set.MapsTo (↑e.symm) e.target e.source
-- PartialHomeomorph.leftInvOn
protected theorem leftInvOn : LeftInvOn e.symm e e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: Set.LeftInvOn (↑e.symm) (↑e) e.source
-- PartialHomeomorph.rightInvOn
protected theorem rightInvOn : RightInvOn e.symm e e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: Set.RightInvOn (↑e.symm) (↑e) e.target
-- PartialHomeomorph.invOn
protected theorem invOn : InvOn e.symm e e.source e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: Set.InvOn (↑e.symm) (↑e) e.source e.target
-- PartialHomeomorph.injOn
protected theorem injOn : InjOn e e.source 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: Set.InjOn (↑e) e.source
-- PartialHomeomorph.bijOn
protected theorem bijOn : BijOn e e.source e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: Set.BijOn (↑e) e.source e.target
-- PartialHomeomorph.surjOn
protected theorem surjOn : SurjOn e e.source e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: Set.SurjOn (↑e) e.source e.target
-- Homeomorph.toPartialHomeomorphOfImageEq_source
@[simps! (config := .asFn) apply symm_apply toPartialEquiv,
  simps! (config := .lemmasOnly) source target]
def _root_.Homeomorph.toPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s)
    (t : Set Y) (h : e '' s = t) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s) (t : Set Y) (h : ⇑e '' s = t)
Goal: (e.toPartialHomeomorphOfImageEq s hs t h).source = s
-- Homeomorph.toPartialHomeomorphOfImageEq_toPartialEquiv
@[simps! (config := .asFn) apply symm_apply toPartialEquiv,
  simps! (config := .lemmasOnly) source target]
def _root_.Homeomorph.toPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s)
    (t : Set Y) (h : e '' s = t) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s) (t : Set Y) (h : ⇑e '' s = t)
Goal: (e.toPartialHomeomorphOfImageEq s hs t h).toPartialEquiv = e.toPartialEquivOfImageEq s t h
-- Homeomorph.toPartialHomeomorphOfImageEq_symm_apply
@[simps! (config := .asFn) apply symm_apply toPartialEquiv,
  simps! (config := .lemmasOnly) source target]
def _root_.Homeomorph.toPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s)
    (t : Set Y) (h : e '' s = t) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s) (t : Set Y) (h : ⇑e '' s = t)
Goal: ↑(e.toPartialHomeomorphOfImageEq s hs t h).symm = ⇑e.symm
-- Homeomorph.toPartialHomeomorphOfImageEq_apply
@[simps! (config := .asFn) apply symm_apply toPartialEquiv,
  simps! (config := .lemmasOnly) source target]
def _root_.Homeomorph.toPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s)
    (t : Set Y) (h : e '' s = t) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s) (t : Set Y) (h : ⇑e '' s = t)
Goal: ↑(e.toPartialHomeomorphOfImageEq s hs t h) = ⇑e
-- Homeomorph.toPartialHomeomorphOfImageEq_target
@[simps! (config := .asFn) apply symm_apply toPartialEquiv,
  simps! (config := .lemmasOnly) source target]
def _root_.Homeomorph.toPartialHomeomorphOfImageEq (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s)
    (t : Set Y) (h : e '' s = t) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y) (s : Set X) (hs : IsOpen s) (t : Set Y) (h : ⇑e '' s = t)
Goal: (e.toPartialHomeomorphOfImageEq s hs t h).target = t
-- Homeomorph.toPartialHomeomorph_source
@[simps! (config := mfld_cfg)]
def _root_.Homeomorph.toPartialHomeomorph (e : X ≃ₜ Y) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y)
Goal: e.toPartialHomeomorph.source = Set.univ
-- Homeomorph.toPartialHomeomorph_apply
@[simps! (config := mfld_cfg)]
def _root_.Homeomorph.toPartialHomeomorph (e : X ≃ₜ Y) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y)
Goal: ↑e.toPartialHomeomorph = ⇑e
-- Homeomorph.toPartialHomeomorph_target
@[simps! (config := mfld_cfg)]
def _root_.Homeomorph.toPartialHomeomorph (e : X ≃ₜ Y) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y)
Goal: e.toPartialHomeomorph.target = Set.univ
-- Homeomorph.toPartialHomeomorph_symm_apply
@[simps! (config := mfld_cfg)]
def _root_.Homeomorph.toPartialHomeomorph (e : X ≃ₜ Y) : PartialHomeomorph X Y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : X ≃ₜ Y)
Goal: ↑e.toPartialHomeomorph.symm = ⇑e.symm
-- PartialHomeomorph.replaceEquiv_eq_self
theorem replaceEquiv_eq_self (e' : PartialEquiv X Y)
    (h : e.toPartialEquiv = e') : e.replaceEquiv e' h = e 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (e' : PartialEquiv X Y) (h : e.toPartialEquiv = e')
Goal: e.replaceEquiv e' h = e
-- PartialHomeomorph.source_preimage_target
theorem source_preimage_target : e.source ⊆ e ⁻¹' e.target 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y)
Goal: e.source ⊆ ↑e ⁻¹' e.target
-- PartialHomeomorph.eq_of_partialEquiv_eq
@[deprecated toPartialEquiv_injective (since := "2023-02-18")]
theorem eq_of_partialEquiv_eq {e e' : PartialHomeomorph X Y}
    (h : e.toPartialEquiv = e'.toPartialEquiv) : e = e' 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] {e e' : PartialHomeomorph X Y} (h : e.toPartialEquiv = e'.toPartialEquiv)
Goal: e = e'
-- PartialHomeomorph.eventually_left_inverse
theorem eventually_left_inverse {x} (hx : x ∈ e.source) :
    ∀ᶠ y in 𝓝 x, e.symm (e y) = y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source)
Goal: ∀ᶠ (y : X) in 𝓝 x, ↑e.symm (↑e y) = y
-- PartialHomeomorph.eventually_left_inverse'
theorem eventually_left_inverse' {x} (hx : x ∈ e.target) :
    ∀ᶠ y in 𝓝 (e.symm x), e.symm (e y) = y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : Y} (hx : x ∈ e.target)
Goal: ∀ᶠ (y : X) in 𝓝 (↑e.symm x), ↑e.symm (↑e y) = y
-- PartialHomeomorph.eventually_right_inverse
theorem eventually_right_inverse {x} (hx : x ∈ e.target) :
    ∀ᶠ y in 𝓝 x, e (e.symm y) = y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : Y} (hx : x ∈ e.target)
Goal: ∀ᶠ (y : Y) in 𝓝 x, ↑e (↑e.symm y) = y
-- PartialHomeomorph.eventually_right_inverse'
theorem eventually_right_inverse' {x} (hx : x ∈ e.source) :
    ∀ᶠ y in 𝓝 (e x), e (e.symm y) = y 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source)
Goal: ∀ᶠ (y : Y) in 𝓝 (↑e x), ↑e (↑e.symm y) = y
-- PartialHomeomorph.eventually_ne_nhdsWithin
theorem eventually_ne_nhdsWithin {x} (hx : x ∈ e.source) :
    ∀ᶠ x' in 𝓝[≠] x, e x' ≠ e x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source)
Goal: ∀ᶠ (x' : X) in 𝓝[≠] x, ↑e x' ≠ ↑e x
-- PartialHomeomorph.nhdsWithin_source_inter
theorem nhdsWithin_source_inter {x} (hx : x ∈ e.source) (s : Set X) : 𝓝[e.source ∩ s] x = 𝓝[s] x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : X} (hx : x ∈ e.source) (s : Set X)
Goal: 𝓝[e.source ∩ s] x = 𝓝[s] x
-- PartialHomeomorph.nhdsWithin_target_inter
theorem nhdsWithin_target_inter {x} (hx : x ∈ e.target) (s : Set Y) : 𝓝[e.target ∩ s] x = 𝓝[s] x 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {x : Y} (hx : x ∈ e.target) (s : Set Y)
Goal: 𝓝[e.target ∩ s] x = 𝓝[s] x
-- PartialHomeomorph.image_eq_target_inter_inv_preimage
theorem image_eq_target_inter_inv_preimage {s : Set X} (h : s ⊆ e.source) :
    e '' s = e.target ∩ e.symm ⁻¹' s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set X} (h : s ⊆ e.source)
Goal: ↑e '' s = e.target ∩ ↑e.symm ⁻¹' s
-- PartialHomeomorph.image_source_inter_eq'
theorem image_source_inter_eq' (s : Set X) : e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X)
Goal: ↑e '' (e.source ∩ s) = e.target ∩ ↑e.symm ⁻¹' s
-- PartialHomeomorph.image_source_inter_eq
theorem image_source_inter_eq (s : Set X) :
    e '' (e.source ∩ s) = e.target ∩ e.symm ⁻¹' (e.source ∩ s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X)
Goal: ↑e '' (e.source ∩ s) = e.target ∩ ↑e.symm ⁻¹' (e.source ∩ s)
-- PartialHomeomorph.symm_image_eq_source_inter_preimage
theorem symm_image_eq_source_inter_preimage {s : Set Y} (h : s ⊆ e.target) :
    e.symm '' s = e.source ∩ e ⁻¹' s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) {s : Set Y} (h : s ⊆ e.target)
Goal: ↑e.symm '' s = e.source ∩ ↑e ⁻¹' s
-- PartialHomeomorph.symm_image_target_inter_eq
theorem symm_image_target_inter_eq (s : Set Y) :
    e.symm '' (e.target ∩ s) = e.source ∩ e ⁻¹' (e.target ∩ s) 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set Y)
Goal: ↑e.symm '' (e.target ∩ s) = e.source ∩ ↑e ⁻¹' (e.target ∩ s)
-- PartialHomeomorph.source_inter_preimage_inv_preimage
theorem source_inter_preimage_inv_preimage (s : Set X) :
    e.source ∩ e ⁻¹' (e.symm ⁻¹' s) = e.source ∩ s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set X)
Goal: e.source ∩ ↑e ⁻¹' (↑e.symm ⁻¹' s) = e.source ∩ s
-- PartialHomeomorph.target_inter_inv_preimage_preimage
theorem target_inter_inv_preimage_preimage (s : Set Y) :
    e.target ∩ e.symm ⁻¹' (e ⁻¹' s) = e.target ∩ s 
Arguments: {X : Type u_1} {Y : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] (e : PartialHomeomorph X Y) (s : Set Y)
Goal: e.target ∩ ↑e.symm ⁻¹' (↑e ⁻¹' s) = e.target ∩ s