ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ModelWithCorners.toHomeomorph_symm_apply
@[simps (config := {simpRhs := true})]
def ModelWithCorners.toHomeomorph {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) [I.Boundaryless] : H ≃ₜ E 
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) [I.Boundaryless]
Goal: ∀ (a : E), I.toHomeomorph.symm a = ↑I.symm a
-- ModelWithCorners.toHomeomorph_apply
@[simps (config := {simpRhs := true})]
def ModelWithCorners.toHomeomorph {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) [I.Boundaryless] : H ≃ₜ E 
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) [I.Boundaryless]
Goal: ∀ (a : H), I.toHomeomorph a = ↑I a
-- contDiffGroupoid_le
theorem contDiffGroupoid_le (h : m ≤ n) : contDiffGroupoid n I ≤ contDiffGroupoid m I 
Arguments: {m n : ℕ∞} {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) (h : m ≤ n)
Goal: contDiffGroupoid n I ≤ contDiffGroupoid m I
-- contDiffGroupoid_zero_eq
theorem contDiffGroupoid_zero_eq : contDiffGroupoid 0 I = continuousGroupoid H 
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)
Goal: contDiffGroupoid 0 I = continuousGroupoid H
-- ofSet_mem_contDiffGroupoid
theorem ofSet_mem_contDiffGroupoid {s : Set H} (hs : IsOpen s) :
    PartialHomeomorph.ofSet s hs ∈ contDiffGroupoid n I 
Arguments: (n : ℕ∞) {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) {s : Set H} (hs : IsOpen s)
Goal: PartialHomeomorph.ofSet s hs ∈ contDiffGroupoid n I
-- symm_trans_mem_contDiffGroupoid
theorem symm_trans_mem_contDiffGroupoid (e : PartialHomeomorph M H) :
    e.symm.trans e ∈ contDiffGroupoid n I 
Arguments: (n : ℕ∞) {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] (e : PartialHomeomorph M H)
Goal: e.symm ≫ₕ e ∈ contDiffGroupoid n I
-- contDiffGroupoid_prod
theorem contDiffGroupoid_prod {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'}
    {e : PartialHomeomorph H H} {e' : PartialHomeomorph H' H'} (he : e ∈ contDiffGroupoid ⊤ I)
    (he' : e' ∈ contDiffGroupoid ⊤ I') : e.prod e' ∈ contDiffGroupoid ⊤ (I.prod I') 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {E' : Type u_5} {H' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] [TopologicalSpace H'] {I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {e : PartialHomeomorph H H} {e' : PartialHomeomorph H' H'} (he : e ∈ contDiffGroupoid ⊤ I) (he' : e' ∈ contDiffGroupoid ⊤ I')
Goal: e.prod e' ∈ contDiffGroupoid ⊤ (I.prod I')
-- SmoothManifoldWithCorners.mk'
theorem SmoothManifoldWithCorners.mk' {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*}
    [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) (M : Type*) [TopologicalSpace M] [ChartedSpace H M]
    [gr : HasGroupoid M (contDiffGroupoid ∞ I)] : SmoothManifoldWithCorners I 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] [gr : HasGroupoid M (contDiffGroupoid ⊤ I)]
Goal: SmoothManifoldWithCorners I M
-- smoothManifoldWithCorners_of_contDiffOn
theorem smoothManifoldWithCorners_of_contDiffOn {𝕜 : Type*} [NontriviallyNormedField 𝕜]
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) (M : Type*) [TopologicalSpace M] [ChartedSpace H M]
    (h : ∀ e e' : PartialHomeomorph M H, e ∈ atlas H M → e' ∈ atlas H M →
      ContDiffOn 𝕜 ⊤ (I ∘ e.symm ≫ₕ e' ∘ I.symm) (I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ range I)) :
    SmoothManifoldWithCorners I 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] (h :
  ∀ (e e' : PartialHomeomorph M H),
    e ∈ atlas H M →
      e' ∈ atlas H M →
        ContDiffOn 𝕜 ⊤ (↑I ∘ ↑(e.symm ≫ₕ e') ∘ ↑I.symm) (↑I.symm ⁻¹' (e.symm ≫ₕ e').source ∩ Set.range ↑I))
Goal: SmoothManifoldWithCorners I M
-- SmoothManifoldWithCorners.subset_maximalAtlas
theorem subset_maximalAtlas [SmoothManifoldWithCorners I M] : atlas H M ⊆ maximalAtlas I 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] [SmoothManifoldWithCorners I M]
Goal: atlas H M ⊆ SmoothManifoldWithCorners.maximalAtlas I M
-- SmoothManifoldWithCorners.chart_mem_maximalAtlas
theorem chart_mem_maximalAtlas [SmoothManifoldWithCorners I M] (x : M) :
    chartAt H x ∈ maximalAtlas I 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] [SmoothManifoldWithCorners I M] (x : M)
Goal: chartAt H x ∈ SmoothManifoldWithCorners.maximalAtlas I M
-- SmoothManifoldWithCorners.compatible_of_mem_maximalAtlas
theorem compatible_of_mem_maximalAtlas {e e' : PartialHomeomorph M H} (he : e ∈ maximalAtlas I M)
    (he' : e' ∈ maximalAtlas I M) : e.symm.trans e' ∈ contDiffGroupoid ∞ 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] {e e' : PartialHomeomorph M H} (he : e ∈ SmoothManifoldWithCorners.maximalAtlas I M) (he' : e' ∈ SmoothManifoldWithCorners.maximalAtlas I M)
Goal: e.symm ≫ₕ e' ∈ contDiffGroupoid ⊤ I
-- PartialHomeomorph.singleton_smoothManifoldWithCorners
theorem PartialHomeomorph.singleton_smoothManifoldWithCorners
    {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)
    {M : Type*} [TopologicalSpace M] (e : PartialHomeomorph M H) (h : e.source = Set.univ) :
    @SmoothManifoldWithCorners 𝕜 _ E _ _ H _ I M _ (e.singletonChartedSpace h) 
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] (e : PartialHomeomorph M H) (h : e.source = Set.univ)
Goal: SmoothManifoldWithCorners I M
-- OpenEmbedding.singleton_smoothManifoldWithCorners
theorem OpenEmbedding.singleton_smoothManifoldWithCorners {𝕜 : Type*} [NontriviallyNormedField 𝕜]
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
    (I : ModelWithCorners 𝕜 E H) {M : Type*} [TopologicalSpace M] [Nonempty M] {f : M → H}
    (h : OpenEmbedding f) :
    @SmoothManifoldWithCorners 𝕜 _ E _ _ H _ I M _ h.singletonChartedSpace 
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] [Nonempty M] {f : M → H} (h : OpenEmbedding f)
Goal: SmoothManifoldWithCorners I M
-- PartialHomeomorph.extend_coe
theorem extend_coe : ⇑(f.extend I) = I ∘ f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: ↑(f.extend I) = ↑I ∘ ↑f
-- PartialHomeomorph.extend_coe_symm
theorem extend_coe_symm : ⇑(f.extend I).symm = f.symm ∘ I.symm 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: ↑(f.extend I).symm = ↑f.symm ∘ ↑I.symm
-- PartialHomeomorph.extend_source
theorem extend_source : (f.extend I).source = f.source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: (f.extend I).source = f.source
-- PartialHomeomorph.isOpen_extend_source
theorem isOpen_extend_source : IsOpen (f.extend I).source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: IsOpen (f.extend I).source
-- PartialHomeomorph.extend_target
theorem extend_target : (f.extend I).target = I.symm ⁻¹' f.target ∩ range I 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: (f.extend I).target = ↑I.symm ⁻¹' f.target ∩ Set.range ↑I
-- PartialHomeomorph.extend_target'
theorem extend_target' : (f.extend I).target = I '' f.target 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: (f.extend I).target = ↑I '' f.target
-- PartialHomeomorph.isOpen_extend_target
lemma isOpen_extend_target [I.Boundaryless] : IsOpen (f.extend I).target 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) [I.Boundaryless]
Goal: IsOpen (f.extend I).target
-- PartialHomeomorph.mapsTo_extend
theorem mapsTo_extend (hs : s ⊆ f.source) :
    MapsTo (f.extend I) s ((f.extend I).symm ⁻¹' s ∩ range I) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set M} (hs : s ⊆ f.source)
Goal: Set.MapsTo (↑(f.extend I)) s (↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I)
-- PartialHomeomorph.extend_left_inv
theorem extend_left_inv {x : M} (hxf : x ∈ f.source) : (f.extend I).symm (f.extend I x) = x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {x : M} (hxf : x ∈ f.source)
Goal: ↑(f.extend I).symm (↑(f.extend I) x) = x
-- PartialHomeomorph.extend_left_inv'
lemma extend_left_inv' (ht : t ⊆ f.source) : ((f.extend I).symm ∘ (f.extend I)) '' t = t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {t : Set M} (ht : t ⊆ f.source)
Goal: ↑(f.extend I).symm ∘ ↑(f.extend I) '' t = t
-- PartialHomeomorph.extend_source_mem_nhds
theorem extend_source_mem_nhds {x : M} (h : x ∈ f.source) : (f.extend I).source ∈ 𝓝 x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {x : M} (h : x ∈ f.source)
Goal: (f.extend I).source ∈ 𝓝 x
-- PartialHomeomorph.extend_source_mem_nhdsWithin
theorem extend_source_mem_nhdsWithin {x : M} (h : x ∈ f.source) : (f.extend I).source ∈ 𝓝[s] x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set M} {x : M} (h : x ∈ f.source)
Goal: (f.extend I).source ∈ 𝓝[s] x
-- PartialHomeomorph.continuousOn_extend
theorem continuousOn_extend : ContinuousOn (f.extend I) (f.extend I).source 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: ContinuousOn (↑(f.extend I)) (f.extend I).source
-- PartialHomeomorph.continuousAt_extend
theorem continuousAt_extend {x : M} (h : x ∈ f.source) : ContinuousAt (f.extend I) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {x : M} (h : x ∈ f.source)
Goal: ContinuousAt (↑(f.extend I)) x
-- PartialHomeomorph.map_extend_nhds
theorem map_extend_nhds {x : M} (hy : x ∈ f.source) :
    map (f.extend I) (𝓝 x) = 𝓝[range I] f.extend I x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {x : M} (hy : x ∈ f.source)
Goal: Filter.map (↑(f.extend I)) (𝓝 x) = 𝓝[Set.range ↑I] ↑(f.extend I) x
-- PartialHomeomorph.map_extend_nhds_of_boundaryless
theorem map_extend_nhds_of_boundaryless [I.Boundaryless] {x : M} (hx : x ∈ f.source) :
    map (f.extend I) (𝓝 x) = 𝓝 (f.extend I x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) [I.Boundaryless] {x : M} (hx : x ∈ f.source)
Goal: Filter.map (↑(f.extend I)) (𝓝 x) = 𝓝 (↑(f.extend I) x)
-- PartialHomeomorph.extend_target_mem_nhdsWithin
theorem extend_target_mem_nhdsWithin {y : M} (hy : y ∈ f.source) :
    (f.extend I).target ∈ 𝓝[range I] f.extend I y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {y : M} (hy : y ∈ f.source)
Goal: (f.extend I).target ∈ 𝓝[Set.range ↑I] ↑(f.extend I) y
-- PartialHomeomorph.extend_image_nhd_mem_nhds_of_boundaryless
theorem extend_image_nhd_mem_nhds_of_boundaryless [I.Boundaryless] {x} (hx : x ∈ f.source)
    {s : Set M} (h : s ∈ 𝓝 x) : (f.extend I) '' s ∈ 𝓝 ((f.extend I) x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) [I.Boundaryless] {x : M} (hx : x ∈ f.source) {s : Set M} (h : s ∈ 𝓝 x)
Goal: ↑(f.extend I) '' s ∈ 𝓝 (↑(f.extend I) x)
-- PartialHomeomorph.extend_target_subset_range
theorem extend_target_subset_range : (f.extend I).target ⊆ range I 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: (f.extend I).target ⊆ Set.range ↑I
-- PartialHomeomorph.interior_extend_target_subset_interior_range
lemma interior_extend_target_subset_interior_range :
    interior (f.extend I).target ⊆ interior (range I) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: interior (f.extend I).target ⊆ interior (Set.range ↑I)
-- PartialHomeomorph.mem_interior_extend_target
lemma mem_interior_extend_target {y : H} (hy : y ∈ f.target)
    (hy' : I y ∈ interior (range I)) : I y ∈ interior (f.extend I).target 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {y : H} (hy : y ∈ f.target) (hy' : ↑I y ∈ interior (Set.range ↑I))
Goal: ↑I y ∈ interior (f.extend I).target
-- PartialHomeomorph.nhdsWithin_extend_target_eq
theorem nhdsWithin_extend_target_eq {y : M} (hy : y ∈ f.source) :
    𝓝[(f.extend I).target] f.extend I y = 𝓝[range I] f.extend I y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {y : M} (hy : y ∈ f.source)
Goal: 𝓝[(f.extend I).target] ↑(f.extend I) y = 𝓝[Set.range ↑I] ↑(f.extend I) y
-- PartialHomeomorph.continuousAt_extend_symm'
theorem continuousAt_extend_symm' {x : E} (h : x ∈ (f.extend I).target) :
    ContinuousAt (f.extend I).symm x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {x : E} (h : x ∈ (f.extend I).target)
Goal: ContinuousAt (↑(f.extend I).symm) x
-- PartialHomeomorph.continuousAt_extend_symm
theorem continuousAt_extend_symm {x : M} (h : x ∈ f.source) :
    ContinuousAt (f.extend I).symm (f.extend I x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {x : M} (h : x ∈ f.source)
Goal: ContinuousAt (↑(f.extend I).symm) (↑(f.extend I) x)
-- PartialHomeomorph.continuousOn_extend_symm
theorem continuousOn_extend_symm : ContinuousOn (f.extend I).symm (f.extend I).target 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H)
Goal: ContinuousOn (↑(f.extend I).symm) (f.extend I).target
-- PartialHomeomorph.extend_symm_continuousWithinAt_comp_right_iff
theorem extend_symm_continuousWithinAt_comp_right_iff {X} [TopologicalSpace X] {g : M → X}
    {s : Set M} {x : M} :
    ContinuousWithinAt (g ∘ (f.extend I).symm) ((f.extend I).symm ⁻¹' s ∩ range I) (f.extend I x) ↔
      ContinuousWithinAt (g ∘ f.symm) (f.symm ⁻¹' s) (f x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {X : Type u_8} [TopologicalSpace X] {g : M → X} {s : Set M} {x : M}
Goal: ContinuousWithinAt (g ∘ ↑(f.extend I).symm) (↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I) (↑(f.extend I) x) ↔
  ContinuousWithinAt (g ∘ ↑f.symm) (↑f.symm ⁻¹' s) (↑f x)
-- PartialHomeomorph.isOpen_extend_preimage'
theorem isOpen_extend_preimage' {s : Set E} (hs : IsOpen s) :
    IsOpen ((f.extend I).source ∩ f.extend I ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set E} (hs : IsOpen s)
Goal: IsOpen ((f.extend I).source ∩ ↑(f.extend I) ⁻¹' s)
-- PartialHomeomorph.isOpen_extend_preimage
theorem isOpen_extend_preimage {s : Set E} (hs : IsOpen s) :
    IsOpen (f.source ∩ f.extend I ⁻¹' s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set E} (hs : IsOpen s)
Goal: IsOpen (f.source ∩ ↑(f.extend I) ⁻¹' s)
-- PartialHomeomorph.map_extend_nhdsWithin_eq_image
theorem map_extend_nhdsWithin_eq_image {y : M} (hy : y ∈ f.source) :
    map (f.extend I) (𝓝[s] y) = 𝓝[f.extend I '' ((f.extend I).source ∩ s)] f.extend I y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set M} {y : M} (hy : y ∈ f.source)
Goal: Filter.map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I) '' ((f.extend I).source ∩ s)] ↑(f.extend I) y
-- PartialHomeomorph.map_extend_nhdsWithin_eq_image_of_subset
theorem map_extend_nhdsWithin_eq_image_of_subset {y : M} (hy : y ∈ f.source) (hs : s ⊆ f.source) :
    map (f.extend I) (𝓝[s] y) = 𝓝[f.extend I '' s] f.extend I y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set M} {y : M} (hy : y ∈ f.source) (hs : s ⊆ f.source)
Goal: Filter.map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I) '' s] ↑(f.extend I) y
-- PartialHomeomorph.map_extend_nhdsWithin
theorem map_extend_nhdsWithin {y : M} (hy : y ∈ f.source) :
    map (f.extend I) (𝓝[s] y) = 𝓝[(f.extend I).symm ⁻¹' s ∩ range I] f.extend I y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set M} {y : M} (hy : y ∈ f.source)
Goal: Filter.map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I] ↑(f.extend I) y
-- PartialHomeomorph.map_extend_symm_nhdsWithin
theorem map_extend_symm_nhdsWithin {y : M} (hy : y ∈ f.source) :
    map (f.extend I).symm (𝓝[(f.extend I).symm ⁻¹' s ∩ range I] f.extend I y) = 𝓝[s] y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {s : Set M} {y : M} (hy : y ∈ f.source)
Goal: Filter.map (↑(f.extend I).symm) (𝓝[↑(f.extend I).symm ⁻¹' s ∩ Set.range ↑I] ↑(f.extend I) y) = 𝓝[s] y
-- PartialHomeomorph.map_extend_symm_nhdsWithin_range
theorem map_extend_symm_nhdsWithin_range {y : M} (hy : y ∈ f.source) :
    map (f.extend I).symm (𝓝[range I] f.extend I y) = 𝓝 y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {M : Type u_3} {H : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [TopologicalSpace H] [TopologicalSpace M] (f : PartialHomeomorph M H) (I : ModelWithCorners 𝕜 E H) {y : M} (hy : y ∈ f.source)
Goal: Filter.map (↑(f.extend I).symm) (𝓝[Set.range ↑I] ↑(f.extend I) y) = 𝓝 y